1 


J 


YX 


y  JJt 


•■■ 


ELEMENTS 


ANALYTICAL    GEOMETRY 


AND    OF    THE 


DIFFERENTIAL  AND  INTEGRAL 


CALCULUS. 


BY  ELIAS   LOOMIS,  LL.D, 

PEOFESSOE   OF   NATCEAL   PHILOSOPHY    AND   ASTRONOMY   IN   TALE   COLLEGE,   AMD    AUTH08 
OF    A  "COURSE   OF   MATHEMATICS. " 


NINETEENTH    EDITION. 


NEW    YORK: 

HARPER   &    BROTHERS,    PUBLISHERS, 
329    &    331    PEARL    STREET 

FRAN  KM  N    SqU  Ali  E. 

1865. 


Entered,  according  to  Act  of  Congress,  in  the  y^ar  1858,  by 

Harper   &   Brothers, 
In  the  Clerk's  Office  of  the  Southern  District  of  New  York. 


PREFACE. 


The  following  treatise  on  Analytical  Geometry  and  the  Cal- 
culus constitutes  the  fourth  volume  of  a  course  of  Mathematics 
designed  for  Colleges  and  High  Schools,  and  is  prepared  upon 
substantially  the  same  model  as  the  preceding  volumes.     It  was 
written,  not  for  mathematicians,  nor  for  those  who  have  a  pe- 
culiar talent  or  fondness  for  the  mathematics,  but  rather  for 
the  mass  of  college  students  of  average  abilities.     I  have,  there- 
fore, labored  to  smooth  down  the  asperities  of  the  road  so  as 
not  to  discourage  travelers  of  moderate  strength  and  courage  ; 
but  have  purposely  left  some  difficulties,  to  arouse  the  energies 
and  strengthen  the  faculties  of  the  beginner.     In  a  course  of 
liberal  education,  the  primary  object  in  studying  the  mathe- 
matics should  be  the  discipline  of  the  mental  powers.     This 
discipline  is  alike  important  to  the  physician  and  the  divine,  the 
jurist  and  the  statesman,  and  it  is  more  effectually  secured 
by  mathematical  studies  than  by  any  other  method  hitherto 
proposed.     Hence  the  mathematics  should  occupy  a  prominent 
place  in  an  education  preparatory  to  either  of  the  learned  pro 
fessions.     But,  in  order  to  secure  the  desired  advantage,  it  is 
indispensable  that  the  student  should  comprehend  the  reasons 
of  the  processes  through  which  he  is  conducted.     How  can  he 
be  expected  to  learn  the  art  of  reasoning  well,  unless  he  see 
clearly  the  foundations  of  the  principles  which  are  taught  ? 
This  remark  applies  to  every  branch  of  mathematical  study, 
but  perhaps  to  none  with  the  same  force  as  to  the  Differential 
and  Integral  Calculus.     The  principles  of  the  Calculus  are  fur- 
ther removed  from  the  elementary  conceptions  of  the  mass  of 
mankind  than  either  Algebra,  Geometry,  or  Trigonometry,  and 
they  require  to  be  developed  with  corresponding  care.     It  is 
quite  possible  for  a  student  to  learn  the  rules  of  the  Calculus, 

B39658 


iv  Preface. 

and  attain  considerable  dexterity  in  applying  them  to  the  solu- 
tion  of  difficult  problems,  without  having  acquired  any  cleai 
idea  of  the  meaning  of  the  terms  Differential  and  Differentia! 
Coefficient.  Cases  of  this  kind  are  not  of  rare  occurrence,  and 
the  evil  may  fairly  be  ascribed,  in  some  degree,  to  the  imper- 
fection of  the  text-books  employed.  The  English  press  lias  foi 
years  teemed  with  "Elementary  treatises  on  the  Calculus,'1 
many  of  which  are  wholly  occupied  with  the  mechanical  pro- 
cesses of  differentiating  and  integrating,  without  any  attempt 
to  explain  the  philosophy  of  these  operations.  A  genuine  math- 
ematician may  work  his  way  through  such  a  labyrinth,  and 
solve  the  difficulties  which  he  encounters  without  foreign  as- 
sistance ;  but  the  majority  of  students,  if  they  make  any  prog- 
ress, will  only  proceed  blindfolded,  and  after  a  time  will  aban- 
don the  study  in  disgust. 

I  have  accordingly  given  special  attention  to  the  develop- 
ment of  the  fundamental  principle  of  the  Differential  Calcu- 
lus, and  shall  feel  a  proportionate  disappointment  if  my  labors 
shall  be  pronounced  abortive.  The  principle  from  which  I  have 
aimed  to  deduce  the  whole  science,  appears  to  me  better  adapt- 
ed to  the  apprehension  of  common  minds  than  any  other ;  and 
although  I  do  not  claim  for  it  any  originality,  it  appears  to  me 
that  I  have  here  developed  it  in  a  more  elementary  manner 
than  I  have  before  seen  it  presented,  except  in  a  small  volume 
by  the  late  Professor  Ritchie,  of  London  University.  I  have  de- 
rived more  important  suggestions  from  this  little  volume,  than 
from  all  the  other  works  on  the  Calculus  which  have  fallen 
under  my  notice.  The  exposition  of  the  principles  of  the  Cal- 
culus contained  in  the  following  treatise,  appears  to  me  so  clear, 
that  I  indulge  the  hope  that  hereafter  this  subject  may  be  made 
a  standard  study  for  all  the  students  of  our  colleges,  and  not 
be  abandoned  entirely  to  the  favored  few. 

While  the  mental  discipline  of  the  majority  of  students  has 
been  the  object  kept  primarily  in  dew,  it  is  believed  that  the 
course  here  pursued  will  be  found  best  adapted  to  develop  the 
taste  of  genuine  mathematicians  ;  for  a  clear  conception  of  the 
fundamental  principles  of  the  science  must  certainly  be  favor- 
able to  future  progress.  The  student  who  renders  himself  fa- 
miliar with  the  present  treatise  will  have  acquired  a  degree  of 


Preface.  v 

mental  discipline  which  will  prove  invaluable  in  every  depart- 
ment of  business ;  and  he  will  be  enabled,  if  so  inclined,  to 
pursue  advantageously  any  of  the  standard  treatises  on  the 
same  subject. 

Every  principle  in  this  work  is  illustrated  by  examples,  and 
at  the  close  of  the  volume  will  be  found  a  large  collection  of 
examples  for  practice,  which  are  to  be  resorted  to  whenever  the 
problems  which  are  incorporated  in  the  body  of  the  work  are 
considered  insufficient. 


CONTENTS. 


ANALYTICAL  GEOMETRY. 
SECTION  I. 

APPLICATION  OF  ALGEBRA  TO  GEOMETRY. 

tagt 

Geometrical  Magnitudes  represented  by  Algebraic  Symbols 9 

Solution  of  Problems *  * 

SECTION  II. 

CONSTRUCTION  OF  EQUATIONS. 

Construction  of  the  Sum  and  Difference  of  two  Quantities 13 

Product  of  several  Quantities " 

Fourth  Proportional  to  three  Quantities 14 

Mean  Proportional  between  two  Quantities    15 

Sum  or  Difference  of  two  Squares 15 

To  inscribe  a  Square  in  a  given  Triangle 16 

To  draw  a  Tangent  to  two  Circles 17 

SECTION  III. 

ON  THE  POINT  AND  STRAIGHT  LINE. 

Methods  of  denoting  the  position  of  a  Point 20 

Abscissa  and  Ordinate  defined - ~0 

Equations  of  a  Point ~1 

Equations  of  a  Point  in  each  of  the  four  Angles 22 

Equation  of  a  straight  Line 23 

Four  Positions  of  the  proposed  Line 24 

Equation  of  the  first  Degree  containing  two  Variables 26 

Equation  of  a  straight  Line  passing  through  a  given  Point 27 

Equation  of  a  straight  Line  passing  through  two  given  Points 28 

Distance  between  two  given  Points 30 

Angle  included  between  two  Lines 30 

Transformation  of  Co-ordinates 32 

Formulas  for  passing  from  one  System  of  Axes  to  a  Parallel  System 33 

Formulas  for  passing  from  Rectangular  Axes  to  Rectangular  Axes 33 

Formulas  for  passing  from  Rectangular  to  Oblique  Axes 34 

Formulas  for  passing  from  Rectangular  to  Polar  Co-ordinates 2s 


viii  Contents. 

SECTION  IV. 

ON  THE  CIRCLE. 

Equation  of  the  Circle  when  the  Origin  is  at  the  Center 36 

Equation  of  the  Circle  when  the  Origin  is  on  the  Circumference 37 

Most  general  form  of  the  Equation 38 

Equation  of  a  Tacgent  to  the  Circle 39 

Polar  Equation  of  the  Circle 42 

SECTION  V. 

ON  THE  PARABOLA. 

Definitions , 4* 

Equation  of  the  Parabola ' 45 

Equation  of  a  Tangent  Line 4G 

Equation  of  a  Normal  Line 48 

The  Normal  bisects  the  Angle  made  by  the  Radius  Vector  and.  Diameter 49 

Perpendicular  from  the  Focus  upon  a  Tangent 50 

Equation  referred  to  a  Tangent  and  Diameter 50 

Parameter  of  any  Diameter 52 

Polar  Equation  of  the  Parabola 53 

Area  of  a  Segment  of  a  Parabola £3 

SECTION  VI. 

ON  THE  ELLirSE. 

Definitions 50 

Equation  of  the  Ellipse  referred  to  its  Center  and  Axes 56 

Equation  when  the  Origiu  is  at  the  Vertex  of  the  Major  Axis CO 

Squares  of  two  Ordinates  as  Products  of  parts  of  Major  Axis 61 

Ordiuates  of  the  Circumscribed  Circle 61 

Every  Diameter  bisected  at  the  Center 62 

Supplementary  Chords „ 63 

Equation  of  a  Tangent  Line 64 

Equation  of  a  Normal  Line 60 

The  Normal  bisects  the  Angle  formed  by  two  Radius  Vectors 67 

Supplementary  Chords  parallel  to  a  Tangent  and  Diameter 68 

Equation  of  Ellipse  referred  to  Conjugate  Diameters 69 

Squares  of  two  Ordinates  as  Products  of  parts  of  a  Diameter 7  i 

Sum  of  Squares  of  two  Conjugate  Diameters 71 

Parallelogram  on  two  Conjugate  Diameters 73 

Polar  Equation  of  the  Ellipse 74 

Area  of  the  Ellipse 75 

SECTION  VII. 

ON  THE  HYPERBOLA. 

Definitions 77 

Equation  of  the  Hyperbola  referred  to  its  Center  and  Axes 78 

Equation  when  the  Origin  is  at  the  Vertex  of  the  Transverse  Axis 81 

Squares  of  two  Ordinates  as  Products  of  parts  of  Transverse  Axis 82 


Contents.  i  x 

Page 

Every  Diameter  bisected  at  the  Center 8C 

Supplementary  Chords 83 

Equation  of  a  Tangent  Lin  3 81 

Equation  of  a  Normal  Line 8.0 

The  Tangent  bisects  the  Angle  contained  by  two  Radius  Vectors 8? 

Supplementary  Chords  parallel  to  a  Tangent  aud  Diameter 88 

Equation  referred  to  Conjugate  Diameters 83 

Squares  of  two  Ordinates  as  the  Rectangles  of  the  Segments  of  a  Diameter. .  91 

Difference  of  Squares  of  Conjugate  Diameters 91 

Parallelogram  on  Conjugate  Diameters 93 

L'olar  Equation  of  the  Hyperbola 93 

Asymptotes  of  the  Hyperbola 94 

Equation  of  the  Hyperbola  referred  to  its  Asymptotes ■. 95 

Parallelogram  contained  by  Co-ordinates  of  the  Curve 90 

Equation  of  Tangent  Line 97 

Portion  of  a  Tangent  between  the  Asymptotes 98 

SECTION  VIII. 

CLASSIFICATION  OF  ALGEBRAIC  CURVES. 

Every  Equation  of  the  second  Degree  is  the  Equation  of  a  Conic  Section  ...  100 

The  Term  containing  the  Product  of  the  Variables  removed 100 

The  Terms  containing  the  first  Tower  of  the  Variables  removed 101 

Lines  divided  into  Classes 101 

Number  of  Lines  of  the  different  orders 10*1 

Family  of  Curves 105 

SECTION  IX. 

TRANSCENDENTAL  CURVE3. 

Cycloid— Defined 106 

Equation  of  the  Cycloid 107 

Logarithmic  Curve — its  Properties 107 

Spiral  of  Archimedes — its  Equation    109 

Hyperbolic  Spiral — its  'Equation 110 

Logarithmic  Spiral— its  Equation Ill 


DIFFERENTIAL  CALCULUS. 

SECTION  I. 

DEFINITIONS  AND  FIRST  PRINCIPLES— DIFFERENTIATION  OF  ALGEBRAIC  FUNC- 
TIONS. 

Definitions — Variables  and  Constants 113 

functions — explicit  and  implicit — increasing  and  decreasing 114 

Limit  of  a  Variable  Quantity 115 

Rate  of  Variation  of  the  Area  of  a  Square 118 

Rate  of  Variation  of  the  Solidity  of  a  Cube 119 

Differential  defined— Differential  Coefficient 1211 


x  Contents. 

Rule  for  finding  the  Differential  Coefficient 122 

Differential  of  any  power  of  a  Variable 123 

Product  of  a  Variable  by  a  Constant   124 

Differential  of  a  Constant  Term 124 

General  expression  for  the  second  State  of  a  Function 125 

Differential  of  the  Sum  or  Difference  of  several  Functions 126 

Differential  of  the  Product  of  several  Functions 127 

Differential  of  a  Fraction 129 

Differential  of  a  Variable  with  any  Exponent 131 

Differential  of  the  Square  Koot  of  a  Variable    133 

Differential  of  a  Polynomial  raised  to  any  Power 133 

SECTION  II. 

SUCCESSIVE  DIFFERENTIALS— MACLAURIN'S  THEOREM— TAYLOR'S   THEOREM 
FUNCTIONS  OF  SEVERAL  INDEPENDENT  VARIABLES. 

Successive  Differentials — Second  Differential  Coefficient 137 

Maclaurin's  Theorem — Applications 138 

Taylor's  Theorem — Applications „ 141 

Differential  Coefficient  of  the  Sum  of  two  Variables 142 

Differentiation  of  Functions  of  two  or  more  independent  Variables 143 

SECTION  III. 

SIGNIFICATION  OF  THE  FIRST  DIFFERENTIAL  COEFFICIENT— MAXIMA  AND  MIN- 
IMA OF  FUNCTIONS. 

Signification  of  the  first  Differential  Coefficient 147 

Maxima  and  Minima  of  Functions  defined 148 

Method  of  finding  Maxima  and  Minima 149 

Application  of  Taylor's  Theorem 151 

How  the  Process  may  be  abridged 154 

Examples 155 

SECTION  IV. 

TRANSCENDENTAL  FUNCTIONS. 

Transcendental  Functions 161 

Differential  of  an  Exponential  Function 161 

Differential  of  a  Logarithm 163 

Circular  Functions 165 

Differentials  of  Sine,  Cosine,  Tangent,  and  Cotangent 166 

Differentials  of  Logarithmic  Sine,  Cosine,  Tangent,  and  Cotangent 170 

Differentials  of  Arc  in  terms  of  its  Sine,  Cosine,  etc 172 

Development  of  the  Sine  and  Cosine  of  an  Arc 174 

SECTION  V. 

APPLICATION  OF  THE  DIFFERENTIAL  CALCULUS  TO  THE  THEORY  OF  CURVES. 

Differential  Equation  of  Lines  of  different  Orders 175 

Length  of  Tangent,  Subtaugent,  Normal,  and  Subnormal 177 

Formulas  applied  to  the  Conic  Sections 178 

Subtangent  of  the  Logarithmic  Curve „ ISO 


Contents.  xi 

P«6» 

Subtangent  and  Tangent  of  Polar  Curves J 82 

Formulas  applied  to  the  Spirals 183 

Differential  of  an  Arc,  Area,  Surface,  and  Solid  of  Revolution    184 

Differential  of  the  Arc  and  Area  of  a  Polar  Curve 190 

Asymptotes  of  Curves 191 

SECTION  VI. 

RADIUS  OF  CURVATURE— EVOLUTES  OF  CURVES. 

Curvature  of  Circles 184 

Radius  of  Curvature  at  any  Point  of  a  Curve 196 

Radius  of  Curvature  of  a  Conic  Section 197 

E  volutes  of  Curves  defined 199 

Equation  of  the  E volute  determined 200 

Evolute  of  the  common  Parabola 201 

Properties  of  the  Cycloid 202 

Expression  for  the  Tangent,  Normal,  etc.,  to  the  Cycloid 202 

Radius  of  Curvature  of  the  Cycloid 203 

Evolute  of  the  Cycloid 204 

SECTION  VII. 

ANALYSIS  OF  CURVE  LINES. 

Singular  Points  of  a  Curve 205 

Tangent  parallel  or  perpendicular  to  Axis  of  Abscissas 205 

When  a  Curve  is  Convex  toward  the  Axis 206 

When  a  Curve  is  Concave  toward  the  Axis   208 

To  determine  a  Point  of  Inflection    209 

To  determine  a  Multiple  Point 211 

To  determine  a  Cusp 21£ 

To  determine  an  isolated  Point    215 


INTEGRAL  CALCULUS. 
SECTION  I. 

INTEGRATION  OF  MONOMIAL  DIFFERENTIALS— OF  BINOMIAL  DIFFERENTIALS- 
OF  THE  DIFFERENTIALS  OF  CIRCULAR  ARCS. 

Integral  Calculus  defined 217 

Integral  of  the  Product  of  a  Differential  by  a  Constant 218 

Integral  of  the  Sum  or  Difference  of  any  number  of  Differentials 219 

Constant  Term  added  to  the  Integral 219 

Integration  of  Monomial  Differentials 219 

Integration  by  Logarithms 220 

Integral  of  a  Polynomial  Differential 221 

Integral  of  a  Binomial  Differential 223 

Definite  Integral 224 

Integrating  between  Limits 225 

Integration  by  Series 226 


xii  Contents. 

Pag< 

Integration  of  the  Differentials  of  Circular  Arcs 227 

Integration  of  Binomial  Differentials 230 

When  a  Binomial  Differential  can  be  integrated 232 

Integration  by  Parts 236 

To  diminish  the  Exponent  of  the  Variable  without  the  Parenthesis 237 

When  the  Exponent  of  the  Variable  is  Negative 242 

To  diminish  the  Exponent  of  the  Parenthesis 243 

When  the  Exponent  of  the  Parenthesis  is  Negative 245 

SECTION  II. 

APPLICATIONS  OF  THE  INTEGRAL  CALCULUS. 

Rectification  of  Plane  Curves 247 

Quadrature  of  Curves 253 

Area  of  Spirals 257 

Area  of  Surfaces  of  Revolution    258 

Oubature  of  Solids  of  Revolution s. 263 


MISCELLANEOUS  EXAMPLES 267 


ANALYTICAL    GEOMETRY. 


SECTION    I. 

APPLICATION  OF  ALGEBRA  TO  GEOMETRV. 

Article  1.  The  relations  of  Geometrical  magnitudes  may 
be  expressed  by  means  of  algebraic  symbols,  and  the  demon- 
strations of  Geometrical  theorems  may  thus  be  exhibited  more 
concisely  than  is  possible  in  ordinary  language.  Indeed,  so 
great  is  the  advantage  in  the  use  of  algebraic  symbols,  that 
they  are  now  employed  to  some  extent  in  all  treatises  on  Ge- 
ometry. 

(2.)  The  algebraic  notation  may  be  employed  with  even 
greater  advantage  in  the  solution  of  Geometrical  problems. 
For  this  purpose  we  first  draw  a  figure  which  represents  all 
the  parts  of  the  problem,  both  those  which  are  given  and  those 
which  are  required  to  be  found.  The  usual  symbols  or  let- 
ters for  known  and  unknown  quantities  are  employed  to  de- 
note both  the  known  and  unknown  parts  of  the  figure,  or  as 
many  of  them  as  may  be  necessary.  We  then  observe  the  re- 
lations which  the  several  parts  of  the  figure  bear  to  each  other, 
from  which,  by  the  aid  of  the  proper  theorems  in  Geometry, 
we  derive  as  many  independent  equations  as  there  arc  un- 
known quantities  employed.  The  solution  of  these  equations 
by  the  ordinary  rules  of  algebra  will  determine  the  value  of 
the  unknown  quantities.  This  method  will  be  illustrated  by  a 
few  examples. 

Ex.  1.  In  a  right-angled  triangle,  having  given  the  hus<-  and 
sum  of  the  hypothenuse  and  perpendicular,  to  find 
the  perpendicular. 

Let  ABC  represent  the  proposed  triangle,  right- 
angled  at  B.  Represent  the  base  A 15  by  l>,  the 
perpendicular  BC  by  x,  and  the  sum  of  the  hypoth- 
snuse  and  perpendicular  by  s;  then  the  hypothe-   A~ 

A 


10 


Analytical   Geometry. 


nuse  will  be  represented  by  s  —  x.      Then,  by  Geom.,  Prop.  11 
B.  IV.,  AT? +13(7= AC2; 

that  is,  b*+x*  =  (s—xy=s2  —  2sx+x\ 

Taking  away  x1  from  each  side  of  the  equation,  we  have 

b"=s2  — 2sx, 

or  2sx=s"~  —  b2 ; 

s*-V 
Whence  x=— - — , 

2s 

from  which  we  see  that  in  any  right-angled  triangle,  the  pei- 

pendicular  is  equal  to  the  square  of  the  sum  of  the  hypothe- 

nuse  and  perpendicular,  diminished  by  the  square  of  the  base, 

and  divided  by  twice  the  sum  of  the  hypothenuse  and  perpen 

dicular.     Thus,  if  the  base  is  3  feet,  and  the  sum  of  the  hy- 

pothenuse  and  perpendicular  9  feet,  the  expression 

92-32 


2s 


comes 


2X9 


=4,  the  perpendicular. 


Ex.  2.  Having  given  the  base  and  altitude  of  any  triangle,  it 
is  required  to  find  the  side  of  the  inscribed  square. 

Let  ABC  represent  the  given  triangle, 
in  which  there  are  given  the  base  AB  and 
the  altitude  CH  ;  it  is  required  to  find  the 
side  of  the  inscribed  square. 

Suppose  the  inscribed  square  DEFG  to 
be  drawn.     Represent  the  base  AB  by  b, 
the  perpendicular  CH  by  h,  and  the  side  of  the  inscribed  square 
by  x  ;  then  will  CI  be  represented  by  'h—x.     Then,  because  GF 
is  parallel  to  the  base  AB,  we  have,  by  similar  triangles,  Geom., 
Prop.  10,  B.  IV., 

AB  :  GF  : :  CH  :  CI ; 
that  is,  b  :x  ::  h  :  h—x; 

or,  since  the  product  of  the  extremes  is  equal  to  that  of  the 

means, 

bh  —  bx=hx ; 

bh 
whence  x~b+h' 

that  is,  the  side  of  the  inscribed  square  is  equal  to  the  product 
of  the  base  and  height  divided  by  their  sum. 

Thus,  if  the  base  of  the  triangle  is  12  feet,  and  the  altitude 
6  feet,  the  side  of  the  inscribed  square  is  found  to  be  4  feet. 


AT  FLIC  AT  ION    Or     A.LGEBHA     TO     GeOMETBY. 


11 


H   E    B 


Ex.  3.  IL/r  •   n  the  base  and  altitudt  of  any  triangle,  it 

is  required  to  inscribe  within  it  <t  rectangle  whose  ride*  shall 
have  to  each  other  a  given  ratio. 

Let  ABC  be  the  given  triangle,  and  sup 
pose  the  required  rectangle  to  be  inscribed 
within  it.  Represent  the  base  AB  by  b, 
the  altitude  CH  by  h,  the  altitude  of  the 
rectangle  DG  by  x,  and  its  base  DE  by  y ; 
also,  let  x  :  y  : :  1  :  n,  or  y=nx. 

Then,  because  the  triangle  GFC  is  similar  to  the  triangle 
ABC,  we  have 

AB  :  GF  : :  CH  :  CI, 
that  is,  b  :  y  : :  h  :  h  —  x  ; 

whence  bh—bx=hy. 

But  since  y=nx,  we  obtain 

bh—bx=hnx. 

Whence  x—j-t — r. 

b+nh 

If  we  suppose  n  equals  unity,  that  is,  the  sides  of  the  rectan- 
gle are  equal  to  each  other,  the  preceding  result  becomes 
identical  with  that  in  Example  2. 

Ex.  4.  The  diameter  of  a  circle  being  given,  to  determine  the 
side  of  the  insc7-ibed  equilateral  triangle. 

Suppose  ABC  to  be  the  required  triangle 
inscribed  in  a  circle  whose  diameter  is  CD. 
Represent  CD  by  d,  and  CB  by  x.  Also 
join  DB.  Theo,  Geom.,  Prop.  15,  Cor.  2, 
B.  III.,  CBD  is  a  right-angled  triangle,  and, 
Geom.,  Prop.  4,  B.  VI.,  BD  is  one  half  of  j£ 
CD. 

Hence  CB'+BD'=CD'; 


that  is, 


Whence 


or 


4 


x  = 


x  =■ 


3(T 
4  ' 

dy/Z 


that  is,  the  side  of  the  inscribed  triangle  i   equal  to  the  dian 
of  the  circle  multiplied  by  half  the  iquare  root  of  three. 


12  Analytical   Geometry. 

Ex.  5.  Given  the  base  b  and  the  difference  d  between  the 
hypothenuse  and  perpendicular  of  a  right-angled  triangle,  to 
find  the  perpendicular. 

Ans.  — . 

2d 

Ex.  6.  Given  the  hypothenuse  h  of  a  right-angled  triangle, 
and  the  ratio  of  the  base  to  the  perpendicular,  as  m  to  n,  to 
find  the  perpendicular. 

nh 

Ans. 


\/m2  +  w* 

Ex.  7.  Given  the  diagonal  d  of  a  rectangle,  and  the  perime- 
ter 4p,  to  find  the  lengths  of  the  sides. 

Ans.p±y—  — p\ 

Ex.  8.  If  the  diagonal  of  a  rectangle  be  10  feet,  and  its  pe- 
rimeter 28  feet,  what  are  the  lengths  of  the  sides  ? 

Ans. 
Ex.  9.  The  diameter,  d,  of  a  circle  being  given,  to  determine 
the  side  of  the  circumscribed  equilateral  triangle. 

Ans.  dy/'S. 
Ex.  10.  From  any  point  within  an  equilateral  triangle,  per- 
pendiculars are  drawn  to  the  three  sides.     It  is  required  to  find 
the  sum,  s,  of  these  perpendiculars. 

Ans.  s=altitude  of  the  triangle. 
Ex.  11.  Given  the  difference,  d,  between  the  diagonal  of  a 
square  and  one  of  its  sides,  to  find  the  length  of  the  sides. 

Ans.  d+d^/2. 
Ex.  12.  In  a  right-angled  triangle,  the  lines  a  and  b,  drawn 
from  the  acute  angles  to  the  middle  of  the  opposite  sides,  are 
given,  to  find  the  lengths  of  the  sides.- 


/4b"-  -a*        ,        /Acf-lr 
Ans.2\/  ——,™&2\/  —^ 


Ex.  13.  Given  the  lengths  of  three  perpendiculars,  a,  b,  and 
c,  drawn  from  a  certain  point  in  an  equilateral  triangle  to  the 
three  sides,  to  find  the  length  of  the  sides. 

,       2(a+b+c) 


SECTION    II. 

CONSTRUCTION  OF  EQUATIONS. 

(3.)  The  construction  of  an  equation  consists  in  finding  a  Geo- 
metrical figure  which  may  be  considered  as  representing  that 
equation ;  that  is,  a  figure  in  which  the  relation  between  the 
parts  shall  be  the  same  as  that  expressed  by  the  equation. 

Problem  I.  To  construct  the  equation  x=a+b. 

The  symbols  a  and  b  being  supposed  to  stand  for  numerical 
quantities  may  be  represented  by  lines.  The  length  of  a  line 
is  determined  by  comparing  it  with  some  known  standard,  as 
an  inch  or  a  foot.  If  the  line  AB  contains  the  standard  unit  a 
times,  then  AB  may  be  taken  to  represent  a.  So,  also,  if  BC 
contains  the  standard  unit  /;  times,  then  BC  may  be  taken  to 
represent  b.     Therefore,  in  order  to  construct  the  expression 

a+b,  draw  an  indefinite  line  AD.     From     i i 

the  point  A  lay  off  a  distance  AB  equal  to 

a,  and  from  B  lay  off"  a  distance  BC  equal  to  b.  then  AC  will 

be  a  right  line  representing  a+b. 

Problem  II.  To  construct  the  equation  x=a  —  b. 

Draw  the  indefinite  line  AD.     From    < 

the  point  A  lay  off  a  distance  AB  equal  to 
a,  and  from  B  lay  off  a  distance  BC  in  the  direction  toward  A 
equal  to  b,  then  will  AC  be  the  difference  between  AB  and 
BC  ;  consequently,  it  may  be  taken  to  represent  the  expres- 
sion a  —  b. 

(4.)  A  single  factor  may  always  be  represented  by  a  lint, 
and  an  algebraic  expression,  consisting  of  a  series  of  letters 
connected  together  by  the  signs  +  and  — ,  may  be  represented 
by  drawing  a  line  of  indefinite  length,  and  setting  off  upon  it  al" 
the  positive  terms  in  one  direction,  and  all  the  negative  terms 
n  the  opposite  direction. 

Problem  III.  To  construct  the  equation  x=ab. 

Let  ABCD  be  a  rectangle  of  which  the  side  AB  contains  tne 
standard  unit  a  times,  and  the  side  AC  contains  the  same  unit 


t4 


Analytical   Geometry. 


D 


E    F 


13 


h  times.  If  we  draw  lines  parallel  to  AC 
through  the  points  E,  F,  etc.,  and  lines 
parallel  to  AB  through  the  points  G,  H  H 
etc.,  the  rectangle  will  be  divided  into  G- 
square  units.  Then,  in  the  first  row, 
AGIB,  there  are  a  square  units ;  in  the 
second  row,  GHKI,  there  are  also  a  square  units,  and  there  are 
as  many  rows  as  there  are  units  in  AC  ;  therefore  the  rectangle 
contains  aXb  square  units,  or  the  rectangle  may  be  consider- 
ed as  representing  the  expression  ab. 

(5.)  The  product  of  two  factors  may,  therefore,  always  be 
represented  by  a  surface. 

Problem  IV.  To  construct  the  equation  x=abc. 

Let  there  be  a  parallelopiped  whose  three  adjacent  edges 
contain  the  standard  unit  respectively  a,  b,  and  c  times ;  then, 
dividing  the  solid  by  planes  parallel  to  its  sides,  we  may  prove 
that  the  number  of  solid  units  in  the  figure  is  aX&Xc,  and,  con- 
sequently, the  parallelopiped  may  be  considered  as  represent- 
ing the  expression  abc. 

(6.)  The  product  of  three  factors  may,  therefore,  always  be 
represented  by  a  solid. 

Problem  V.  To  construct  the  equation  x= — . 

c 


If        ah 
If  x— — ,  then 
c 


b  :  x  ; 


that  is,  x  is  a  fourth  proportional  to  the  three  given  quantities, 
c,  a,  and  b ;  hence  the  line  whose  length  is  expressed  by  x  is 
a  fourth  proportional  to  three  lines  whose  respective  lengths 
are  c,  a,  and  b. 

From  A  draw  two  lines  AB,  AC  mak- 
ing any  angle  with  each  other.  From  A 
lay  off  a  distance  AD  equal  to  c,  AB  equal 
to  a,  and  AE  equal  to  b.  Join  DE,  and 
through  B  draw  BC  parallel  to  DE ;  then 
will  AC  be  equal  to  x. 

For,  by  similar  triangles,  we  have 

AD  :  AB  : :  AE  :  AC, 
or  c    \    a    '.'.    b    :  AC. 

ab 


Hence 


AC=- 


Construction   of    EaaATioNS. 


15 


Problem  VI.   To  construct  the  equation  x=— =-. 

de 

This  expression  can  be  put  under  the  form 

abXc  ab     c 

dXe  '  d     e 

First  find  a  fourth  proportional  m  to  the  three  quantities  d,  a, 

and  b ;  that  is,  make 

d  :  a  ::b  :  m ;  whence  m=-—. 


d 


The  proposed  expression  then  becomes 

mc 


which  may  be  constructed  as  in  Problem  V. 

Problem  VII.  To  construct  the  equation  x=  Vab. 

Since  Vab  is  a  mean  proportional  between  a  and  b,  the 
problem  requires  us  to  find  geometrically  a  mean  proportional 
between  a  and  b. 

Draw  an  indefinite  straight  line,  and  upon  p, 

it  set  off  AB  equal  to  a,  and  BC  equal  to  b. 
On  AC,  as' a  diameter,  describe  a  semicir- 
cle, and  from  B  draw  BD  perpendicular  to 
AC,  meeting  the  circumference  in  D  ;  then  A  B        c 

BD  is  a  mean  proportional  between  AB  and  BC  (Geom.,  Prop 
22,  Cor.,  B.  IV.).  Hence  BD  is  a  line  representing  the  expres- 
sion Vab. 

Problem  VIII.  To  construct  the  equation  x—  Va2  +  if. 

Draw  the  line  AB,  and  make  it  equal  to  a  ; 
from  B  draw  BC  perpendicular  to  AB,  and 
make  it  equal  to  b.  Join  AC,  and  it  will 
represent  the  value  of  V c?  +  b'\  For  AC2= 
AB2+BC2  (Geom.,  Prop.  11,  B.  IV.). 

Problem  IX.  To  construct  the  equation  x=  Va*—b 

Draw  an  indefinite  right  line  AB ;  at  B 
draw  BC  perpendicular  to  AB,  and  make  it 
equal  to  b.  With  C  as  a  center,  and  a  radius 
equal  to  a,  describe  an  arc  of  a  circle  cut- 
.'ing  AB  in  D;  then  will  BD  represent  the  A     D\  B 

expression  Va'  —  b*.     For 

BD^DC'-BC^a2-^; 
Whence  BD  =  vV-69. 


It> 


Analytical    Geometry. 


Problem  X.  To  construct  the  equation  x=a^Vai—bt. 

Draw  an  indefinite  line  AE,  and  set  c 

off  a  distance  AB  equal  to  a;  from  B 
draw  BC  perpendicular  to  AB,  and  make 
it  equal  to  b.     With  C  as  a  center,  and 


a  radius  equal  to  a,  describe  an  arc  of  a   A    D"- -"  E 

circle  cutting  AE  in  D  and  E.  Now  the  value  of  Vai—b 
will  be  BD  or  BE.  When  the  radical  is  positive,  its  value  is 
to  be  set  off  toward  the  right ;  when  negative,  toward  the  left. 
Therefore,  AD  and  AE  are  the  values  required ;  for 

AE  =  AB+BE=a+  vV^F, 
and  AD=AB-BD=«-vV-&\ 

The  preceding  values  are  the  roots  of  the  equation 
x*-2ax=-b*. 

Problem  XI.  Having  given  the  base  and  altitude  of  any 
triangle,  it  is  required  to  find  the  side  of  the  inscribed  square 
by  a  geometrical  construction. 

We  have  found  on  page  10  the  side  of  the  inscribed  square  to 

bh 
be  equal  to  .  ,  .. 
1  b+h 

Hence  the  side  of  the  inscriDed  square  is  a  fourth  propor- 
tional to  the  three  lines  b+h,  b,  and  h. 

Produce  the  side  CA  indefinite-  c 

ly,  and  lay  offCK  equal  to  the  al- 
titude //,  and  KL  equal  to  the  base 
b.  Join  LH,  and  draw  KI  paral- 
lel to  LH  ;  IH  will  be  equal  to  a 
side  of  the  inscribed  square. 

For,  by  similar   triangles,   we 
have        "  CL  :  KL  : :  CH  :  IH  ;         /. 
that  is,       b+h  :    b     :  :    h    :  IH.     L-:""" 

bh 


-"'H 


Hence 


IH= 


b+li 


and  therefore  IH  is  equal  to  a  side  of  the  inscribed  square. 

Example  3,  page  11,  maybe  constructed  in  similar  manner 
oy  laying  off  CK  equal  to  nh. 

Problem  XII.  It  is  required  to  draw  a  common  tangent  line 
to  two  given  circles  situated  in  the  same  plane. 

LetCC  be  the  centers  of  the  two  circles,  CM,  CM'  their  radii 


Construction   of   Equations. 


17 


Let  us  suppose  the  problem 
solved,  and  that  MM'  is  the 
common  tangent  line.  Pro- 
duce this  tangent  until  it  meets 
the  line  CC,  passing  through 
the  centers  of  the  circles ;  then, 
drawing  the  radii  CM,  CM'  to  the  points  of  tangency,  the  an- 
gles CMT,  C'M'T  will  be  right  angles,  and  the  triangles  CMT, 
G'M'T  will  be  similar.     Hence  we  shall  have  the  proportion 

CM  :  CM'  : :  CT  :  CT. 

Represent  CM  by  r,  CM'  by  r ,  CC  by  a,  and  CT  by  x.     CT 
will,  therefore,  be  x— a,  and  the  preceding  proportion  will  be- 
come r  '.  r' : :  x  :  x  —  a ; 
whence  rx—ra=r'x, 

ar 

and  x= -; 

r—r 

from  which  we  see  that  CT  or  a;  is  a  fourth  proportional  to  the 

three  lines  r—r1,  a,  and  r. 

To  obtain  a:  by  a  geometrical  construction,  through  the  cen- 
ters C,  C  draw  two  parallel  radii,  CN,  C'N'.  Through  N  and 
N'  draw  the  line  NN'T,  meeting  the  line  CC  in  T.  Through 
T  draw  a  tangent  line  to  one  of  the  circles,  it  will  also  be  a 
tangent  to  the  other. 

For  through  N'  draw  N'D 
parallel  to  CT ;  then  N'D  will 
represent  a,  ND  will  repre- 
sent r—r1 ;  and,  since  the 
triangle  DNN'  is  similar  to 
CNT,  we  have  the  proportion 

DN:DN 
or  r—r'  :     a 


whence 


CT 


which  is  the  value  of  a;  before  found.  Therefore,  a  line  drawn 
from  T,  tangent  to  one  of  the  circles,  will  also  be  tangent  to 
the  other  ;  and,  since  two  tangent  lines  can  be  drawn  from  the 
point  T,  we  see  that  the  problem  proposed  admits  of  two  solu- 
tions. 

Cor.  1.  If  we  suppose  the  radius  r  of  the  first  circle  to  remain 

B 


18 


Analytical  Geometry. 


constant,  and  the  smaller  radius  r'  to  increase,  the  difference 
r— r'  will  diminish;  and,  since  the  numerator  ar  remains  con- 
stant, the  value  of  a:  will  increase  ;  which  shows  that  the  nearer 
the  two  circles  approach  to  equality,  the  more  distant  is  the 
point  of  intersection  of  the  tangent  line  with  the  line  joining  the 
centers.  When  the  two  radii  r  and  r'  become  equal,  the  de- 
nominator becomes  0,  and  the  value  of  ar  becomes  infinite. 

Cor.  2.  If  we  suppose  r  to  increase  so  as  to  become  greater 
than  r,  the  value  of  a:  becomes  negative,  which  shows  that  the 
point  T  falls  to  the  left  of  the  two  circles. 

Cor.  3.  Two  other  tangent  lines  may  be  drawn  intersecting 
each  other  between  the  circles.     If 
we  represent  CT  by  x,  the  radii  of 
the  circles  by  r  and  r',  and  the  dis- 
tance between  their  centers  by  a, 
we  shall  have  from  the  similar  tri- 
angles CMT,  C'M'T,  the  proportion 
CM  :  CM'  : :  CT 
or  r  :     r'     : :    x 

ar 
whence  x=- 


CT, 

a—x 


r+r' 


This  expression  may  be  constructed  in  a  manner  similar  to 
the  former.  Through  the  centers 
C  and  C  draw  two  parallel  radii 
CN,  CN',  lying  on  different  sides  / 
of  the  line  CC;  join  the  points  NN', 
and  through  T,  where  this  line  in- 
tersects CC,  draw  a  line  tangent  to 
one  of  the  circles,  it  will  be  a  tangent  to  the  other.  For 
through  N'  draw  N'D  parallel  to  CC,  and  meeting  CN  pro- 
duced in  D.  DN'  will  then  represent  a,  ND  will  represent 
r+r',  and  the  similar  triangles  NCT,  NDN'  will  furnish  the 

proportion 

ND  :  DN'  : :  NC  :  CT, 

or  r+r'  :     a     : :    r    :  CT; 

ar 


whence 


CT: 


r+r 
which  is  the  value  already  found  for  x. 

(7.)  Every  Algebraic  expression,  admitting  of  geometrical 
construction,  must  hare   all  its  terms  homogeneous  (Algebra 


Construction   of   Equations.  19 

A.rt.  31)  ;  that  is,  each  term  must  contain  the  same  number  of 
literal  factors.  Thus,  each  term  must  either  be  of  one  dimen* 
sion,  and  so  represent  a  line ;  or,  secondly,  each  must  be  of 
two  dimensions,  and  represent  a  surface  ;  or,  thirdly,  each  must 
be  of  three  dimensions,  and  denote  a  solid ;  since  dissimilar 
geometrical  magnitudes  can  neither  be  added  together  nor  sub- 
tracted from  each  other. 

It  may,  however,  happen  that  an  expression  really  admitting 
of  geometrical  construction  appears  to  be  not  homogeneous ; 
but  this  result  arises  from  the  circumstance  that  the  geometrical 
unit  of  length,  being  represented  algebraically  by  1,  disappears 
from  all  algebraic  expressions  in  which  it  is  either  a  factor  or 
a  divisor.  To  render  these  results  homogeneous,  it  is  only 
necessary  to  restore  this  divisor  or  factor  which  represents 
unity. 

Thus,  suppose  we  have  an  equation  of  the  form 

x=ab+c. 

If  we  put  I  to  represent  the  unit  of  measure  for  lines,  we  may 

change  it  into  the  homogeneous  equation, 

lx=ab+cl, 

ab 
or  x=—+c ; 

which  is  easily  constructed  geometrically. 


SECTION   III. 

ON  THE  POINT  AND  STRAIGHT  LINE. 
(8.)  There  are  two  methods  of  denoting  the  position  of  a 
point  in  a  plane.     The  first  is  by  means  of  the  distance  and 
direction  of  the  proposed  point  from  a  given 
point.     Thus,  if  A  be  a  known  point,  and 
AX  be  a  known  direction,  the  position  of 
the  point  P  will  be  determined  when  we  a 
know  the  distance  AP  and  the  angle  PAX. 

The  assumed  point  A  is  called  the  pole ;  the  distance  of  P 
from  A  is  called  the  radius  vector ;  and  the  radius  vector,  to 
aether  with  its  angle  of  inclination  to  the  fixed  line,  are  called 
the  polar  co-ordinates  of  the  point. 

(9.)  It  is,  however,  generally  most  convenient  to  denote  the 
position  of  a  point  by  means  of  its  distances  from  two  given 
lines  which  intersect  one  another.  Thus, 
let  AX,  AY  be  two  assumed  straight  lines 
which  intersect  in  any  angle  at  A,  and  let  P 
be  a  point  in  the  same  plane ;  then,  if  we 
draw  PB  parallel  to  AY,  and  PC  parallel  to 
AX,  the  position  of  the  point  P  will  be  de-   A.  13 

noted  by  means  of  the  distances  PB  and  PC. 

The  two  lines  AX,  AY,  to  which  the  position  of  the  point  P 
is  referred,  are  called  axes,  and  their  point  of  intersection  A  is 
called  their  origin.  The  distance  AB,  or  its  equal  CP,  is  called 
the  abscissa  of  the  point  P ;  and  BP,  or  its  equal  AC,  is  called 
the  ordinate  of  the  same  point.  Hence  the  axis  AX  is  called 
the  axis  of  abscissas,  and  AY  is  called  the  axis  of  ordinates. 

The  abscissa  and  ordinate  of  a  point,  when  spoken  of  to- 
gether, are  called  the  co-ordinates  of  the  point,  and  the  two 
axes  are  called  co-ordinate  axes. 

The  axes  are  called  oblique  or  rectangular,  according  as 
YAX  is  an  oblique  or  a  right  angle.  Rectangular  axes  are  the 
most  simple,  and  will  generally  be  employed  in  this  treatise. 


On   the  Point   and   straight   Line.  2i 

An  abscissa  is  generally  denoted  by  the  letter  x,  and  an  or- 
dinate by  the  letter  y ;  and  hence  the  axis  of  abscissas  is  often 
called  the  axis  of  X,  and  tne  axis  of  ordinates  the  axis  of  Y. 

The  abscissa  of  any  point  is  its  distance  from  the  axis  of  or- 
dinates, measured  on  a  line  parallel  to  the  axis  of  abscissas. 

The  ordinate  of  any  point  is  its  distince  from  the  axis  of  ab- 
scissas, measured  on  a  line  parallel  to  ,lhe  axis  of  ordinates. 

(10.)  The  position  of  a  point  may  be  determined  when  its 
co-ordinates  are  known.  For  suppose  the  abscissa  of  the  point 
P  is  equal  to  a,  and  its  ordinate  is  equal  to  b. 
Then,  to  determine  the  position  of  the  point 
P,  from  the  origin  A  lay  off  on  the  axis  of 
abscissas   a  distance   AB   equal  to   a,  and       J  j 

through  B  draw  a  line  parallel  to  the  axis      f  /       ^ 

of  ordinates.     On  this  line  lay  off  a  distance  -^  B 

BP  equal  to  b,  and  P  will  be  the  point  required. 

Hence,  in  order  to  determine  the  position  of  a  point,  we  need 
only  have  the  two  equations 

x=a,  y=b, 
in  which  a  and  b  are  given.     These  equations  are,  therefore, 
nailed  the  equations  of  a  point. 

(11.)  It  is,  however,  necessary,  in  order  to  determine  the 
position  of  a  point,  that  not  only  the  absolute  values  of  a  and  b 
should  be  given,  but  also  the  signs  of  these  quantities.  If  the 
axes  are  produced  through  the  origin  Y 

to  X'  and  Y',  it  is  obvious  that  the  ,  /         p 

abscissas   reckoned  in  the  direction  I         7        7 

AX'  ought  not  to  have  the  same  sign  X- 
as  those  reckoned  in  the  opposite  di- 
rection AX ;  nor  should  the  ordinates 
measured  in  the  direction  AY'  have 
the  same  sign  as  those  measured  in  the  opposite  direction  AY  ; 
for  if  there  were  no  distinction  in  this  respect,  the  position  of  a 
point  as  determined  by  its  equations  would  be  ambiguous. 
Thus  the  equations  of  the  point  P  would  equally  belong  to  the 
points  P',  P",  P"',  provided  the  absolute  lengths  of  the  co-or- 
dinates of  each  were  equal  to  those  of  P.  All  this  ambiguity 
is  avoided  by  regarding  the  co-ordinates  which  are  measured 
in  one  direction  as  plus,  and  those  in  the  opposite  direction 
minus.     It  is  generally  agreed  to  regard  those  abscissas  which 


22  Analytical   Geometry. 

fall  on  the  right  of  the  origin  A  as  positive,  and  hence  those 
which  fall  on  the  left  must  be  considered  negative.  So,  also, 
it  has  been  agreed  to  consider  those  ordinates  which  are  above 
the  origin  as  positive,  and  hence  those  which  fall  below  it  must 
be  considered  negative. 

(12.)  The  angle  YAX  is  called  the  first  angle;  YAX'  the 
second  angle ;  Y'AX'  the  third  angle ;  and  Y'AX  the  fourth 
angle. 

The  following,  therefore,  are  the  equations  of  a  point  in  each 
of  the  four  angles  : 

For  the  point  P     in  the  first  angle,       x=+a,  y=+b, 
"  P'        "      second  angle,  x=  —  a,  y=+b, 

"  P"       "      third  angle,     x=—a,y=-b, 

P'"      "      fourth  angle,   x=+a,  y=-b. 
If  the  point  be  situated  on  the  axis  AX,  the  equation  y=b 
becomes  y=0,  so  that  the  equations 
x=*±a,  y=0, 

characterize  a  point  on  the  axis  of  abscissas  at  the  distance  a 
from  the  origin. 

If  the  point  be  situated  on  the  axis  AY,  the  equation  x=a 
becomes  x=0,  so  that  the  equations 

x—0,  y=±b, 
characterize  a  point  on  the  axis  of  ordinates  at  the  distance  b 
from  the  origin. 

If  the  point  be  common  to  both  axes,  that  is,  if  it  be  at  the 
origin,  its  position  will  be  expressed  by  the  equations 

x=0,  y=0. 

Ex.  1.  Determine  the  point  whose  equations  are  x=+4, 
y=-3. 

Ex.  2.   Determine  the  point  whose  equations  are  x=  —  2 

Ex.  3.  Determine  the  point  whose  equations  are  x—0 
y——b. 

Ex.  4.  Determine  the  point  whose  equations  are  x=—  8, 
y=0. 

Definition. — The  equation  of  a  line  is  the  equation  which  ex- 
presses the  relation  between  the  two  co-ordinates  of  every  point  of 
the  line. 


On   the   Point   and  straight   Lin 


23 


Proposition  I. — Theorem. 

(13.)   TJie  equation,  of  a  straight  line  referred  to  rectangular 
axes  is 

y=ax-\-b ; 
where  x  and  y  are  the  co-ordinates  of  any  point  of  the  line,  a 
represents  the  tangent  of  the  angle  which  the  line  makes  with 
the  axis  of  abscissas,  and  b  the  distance  from  the  origin  at 
which  it  intersects  the  axis  of  ordinates.  Also,  a  and  b  may 
be  either  positive  or  negative. 

Let  A  be  the  origin  of  co-ordinates, 
AX  and  AY  be  rectangular  axes,  and 
PC  any  straight  line  whose  equation 
is  required  to  be  determined.  Take 
any  point  P  in  the  given  line,  and 
draw  PB  perpendicular  to  AX  ;  then  -^ 
will  PB  be  the  ordinate,  and  AB  the  abscissa  of  the  point  P. 
From  A  draw  AD  parallel  to  CP,  meeting  the  line  BP  in  D. 

Let  AB=£, 

BP=y, 
tangent  PEX  or  DAX=a, 
and  AC  or  DP=6. 

Then,  by  Trigonometry,  Theorem  II.,  Art.  42, 
R  :  AB  : :  tang.  DAX  :  BD, 
or  R  :    x    : :  a  :  BD. 

Hence,  calling  the  radius  unity,  we  have 

BT)=ax. 
But  BP=BD+DP; 

hence  y=  ax  +b. 

If  the  line  CP  cuts  the  axis  of  ordi- 
nates below  the  origin,  then  we  shall 
have  BP=BD-DP, 

or  y=  ax  —b. 

If  x  represents  a  negative  line,  as 
AB,  then  will  ax  or  BD  be  negative, 
and  the  equation  y  =  ax+b  will  be  true, 
since  BP=-BD  +  DP. 

(14.)  The  line  PC  has  been  drawn  - 
so  as  to  make  an  acute  angle  with  the 
axis  of  abscissas;   but  the  preceding 


24 


Analytical   Geometry. 


equation  is  equally  applicable  whatever  may  be  this  angle, 
provided  proper  signs  are  attributed  to  each  term.  The  angle 
which  the  line  makes  with,  the  axis  of  abscissas  is  supposed 
to  be  measured  from  the  axis  AX  around  the  circle  by  the 
left. 

If  the  angle  is  obtuse,  its  tangent  will  be  negative  (Trigo- 
nometry, Art.  70).     Thus,  if  PC  be  the  po- 
sition of  the  proposed  line  with  reference  to 
the  rectangular  axes  AX,  AY,  then,  in  the 
proportion 

E:AB::tang.  DAX:BD, 
the  tangent  of  DAX  is  negative;  BD  is 
therefore  negative.     The  equation  of  this 
line  may  then  be  written 

y=—ax-\-b, 
where  it  must  be  observed  that  the  sign  —  applies  only  to  the 
quantity  a,  and  not  to  a?,  for  the  sign  of  x  depends  upon  its  di- 
rection from  the  origin  A. 

(15.)  There  may,  therefore,  be  four  positions  of  the  pro- 
posed line,  and  these  positions  are  indicated  by  the  signs  of  a 
and  b  in  the  general  equation. 

1.  Let  the  line  take  the  position  shown 
in  the  annexed  diagram,  cutting  the  axis 
of  X  to  the  left  of  the  origin,  and  the 
axis  of  Y  above  it,  then  a  and  b  are  both 
positive,  and  the  equation  is 

y=+ax+b. 

2.  If  the  line  cuts  the  axis  of  X  to  the 
right  of  the  origin,  and  the  axis  of  Y  be- 
low it,  then  a  will  still  be  positive,  but  b 
will  be  negative,  and  the  equation  be- 
comes 

y=+ax— b. 

3.  If  the  line  cuts  the  axis  of  X  to  the 
right  of  the  origin,  and  the  axis  of  Y 
above  it,  then  a  becomes  negative  and  b 
positive.  In  this  case,  therefore,  the 
equation  is 

y~—  ax+b. 

4.  If  the  line  cuts  the  axis  of  X  to  the  left  of  the  origin,  and 


On   the  Point   and  straight   Line. 


25 


.he  axis  of  Y  below  it,  then  both  a  and  b 
will  be  negative,  so  that  the  equation  be- 
comes y—  —  ax—b. 

If  we  suppose  the  straight  line  to  pass 
through  the  origin  A,  then  b  will  be  equal 
to  zero,  and  the  general  equation  becomes 

y=ax, 
which  is  the  equation  of  a  straight  line  passing  through  the 

origin. 

We  here  suppose  the  letters  a  and  b  to  stand  for  positive 
quantities.  It  is,  however,  to  be  borne  in  mind  that  they  may 
themselves  represent  negative  quantities,  in  which  case  —a 
and  —b  will  be  positive. 

Ex.  1.  Let  it  be  required  to  draw  the  line  whose  equation  is 
y=2x+4. 

If  in  this  equation  we  make  x=0,  the  value  of  y  will  designate 
the  point  in  which  the  line  intersects  the  axis  of  ordinates,  for 
that  is  the  only  point  of  the  line  whose  abscissa  is  0.  This 
supposition  will  give 

y=4. 

Having  drawn  the  co-ordinate  axes  AX, 
AY,  lay  off  from  the  origin  A  a  distance 
AB  equal  to  4 ;  this  will  be  one  point  of 
the  required  line. 

Again,  if  in  the  proposed  equation  we 
make  y=0,  the  value  of  x,  which  is  found 
from  the  equation,  will  designate  the  point 
in  which  the  line  intersects  the  axis  of  abscissas,  for  that  is  tne 
only  point  of  the  line  whose  ordinate  is  0.  This  supposition 
will  give  2x=—  4, 

or  x=  —  2. 

Lay  off  from  the  origin  A,  toward  the  left,  a  distance  AC 
equal  to  2  ;  this  will  give  a  second  point  of  the  proposed  line, 
and  the  line  may  be  drawn  through  the  two  points  B  and  C. 

(1G.)  We  may  determine  any  number  of  points  in  this  line 
by  assuming  particular  values  for  x  or  y ;  the  equation  will 
furnish  the  corresponding  value  of  the  other  variable. 

Making  successively 

£C=1,  we  find  y=6,  x=3,  we  find  y=l0, 

x=2,       "       y  =  8,  x=4,       "       y= 12,  etc. 

In  order  to  represent  these  values  by  a  figure,  we  draw  two 


H 
G 


26  Analytical  Geo METRr. 

axes  AX,  AY  at  right  angles  to  each  other 
Then,  in  order  to  construct  the  values  x—1, 
y=6,  we  set  off  on  the  axis  of  abscissas  a  -y  i 
line  AB  equal  to  1,  and  erect  a  perpendicu- 
lar BG  equal  to  6  ;  this  determines  one 
point  of  the  required  line.  Again,  take 
AC  equal  to  2,  and  make  the  perpendicular 
CH  equal  to  8 ;  this  will  determine  a  sec- 
ond point  of  the  required  line.  In  the  same  BCI)E 
manner  we  may  determine  the  points  K  and  L,  and  any  num- 
ber of  points.  The  required  line  must  pass  through  all  the 
points,  G,  H,  K,  L,  etc. 

Any  straight  line  may  be  constructed  by  determining  two 
points  in  that  line,  and  drawing  the  line  through  those  points. 

Ex.  2.  Construct  the  line  whose  equation  is  y=2x+3. 

Ex.  3.  Construct  the  line  whose  equation  is  y=Sx— 7. 

Ex.  4.  Construct  the  line  whose  equation  is  y—  —  x-\-2. 

Ex.  5.  Construct  the  line  whose  equation  is  y=—  2x— 5. 

Ex.  6.  Construct  the  line  whose  equation  is  y=ox. 

Ex.  7.  Construct  the  line  whose  equation  is  y—b. 

Ex.  8.  Construct  the  line  whose  equation  is  y=—2. 

In  the  equation  y=ax+b,  the  quantities  a  and  b  remain  the 
same,  while  the  co-ordinates  x  and  y  vary  in  value  for  every 
point  in  the  same  line.  We,  therefore,  call  a  and  b  constant 
quantities,  and  x  and  y  variable  quantites. 

Proposition  II. — Theorem. 
(17.)  Every  equation  of  the  first  degree  containing  two  varia- 
bles is  the  equation  of  a  straight  line. 

Every  equation  of  the  first  degree  containing  two  variables 
can  be  reduced  to  the  form 

Ay=Bx+C, 
in  which.  A,  B,  and  C  may  be  positive  or  negative. 

Now  a  straight  line  may  always  be  constructed  of  which 
this  shall  be  the  equation. 

Draw  the  co-ordinate  axes  AX,  AY  at  right  angles  to  each 


On  the  Point  and  straight  Line. 


27 


other;  make  AB  equal  to  C^A,  and  AC 
equal  to  C-hB,  and  through  the  points  B 
and  C  draw  the  line  PBC,  it  will  be  the 
required  line. 

For  the  equation  of  this  line  is 

AB    ,   ._ 

but  by  the  construction, 

AB_C_C_B. 

AO~A    B~A' 
also,  AB=C-A. 

Therefore  the  equation  of  the  line  PBC  is 

B      C 


X      A 


(2) 


or, 


^  =  A*+A; 
A*/=Ba;  +  C. 


Examples.— Draw  the  lines  of  which  the  following  are  the 
equations, 

2y=3x-5,  2x=y+7,  cc+y=0, 

y=±-x,  x=2y,  a=4, 

x+y=lO,  x+y±lO  =  0,       y=2. 

What  are  the  values  of  a  and  b  in  these  equations? 

Proposition  III. — Theorem. 
(18.)  The  equation  of  a  straight  line  passing  through  a  given 
point  is 

y-y'  =  a{x-x'\ 
where  x'  and  y'  denote  the  co-ordinates  of  the  given  point, 
x  and  y  the  co-ordinates  of  any  point  of  the  line,  and  a  the 
tangent  of  the  angle  which  the  line  makes  with  the  axis  of 
abscissas. 

Known  co-ordinates  are  frequently  designated  by  marking 
them  thus, 

x',y';  x",y";  x"',y"',etc, 
which  are  read  x  prime,  y  prime;  x  second,  y  second;  a;  third, 
y  third,  etc.  T 

Let  P  be  the  given  point,  and  designate 
its  co-ordinates  by  x'  and  y'.  Then,  since 
the  general  equation  for  every  point  in  the 
required  line  is 

y  =  ax+b;  (1) 


Analytical    Geometry. 


and  since  P  is  a  point  in  the  line,  it  follows  that 

y'  =  ax'  +  b.  (2) 

By  means  of  equation  (2)  we  may  eliminate  b  from  equa- 
tion (1). 

Subtracting  equation  (2)  from  equation  (1),  we  obtain 
y-y'=a(x-x'), 
which  is  the  equation  of  a  line  passing  through  the  given 
point  P. 

Since  the  tangent  a,  which  fixes  the  direction  of  the  line,  is 
not  determined,  there  may  be  an  infinite  number  of  straight 
lines  drawn  through  a  given  point.  This  is  also  apparent  from 
the  figure. 

(19.)  If  it  be  required  that  the  line  shall  pass  through  a  given 
point,  and  be  parallel  to  a  given  line,  then  the  angle  which  the 
line  makes  with  the  axis  of  abscissas  is  determined ;  and  if 
we  put  a'  for  the  tangent  of  this  angle,  the  equation  of  the  line 
sought  will  be 

y— y'=a'{x— x'). 

Ex.  Draw  a  line  through  the  point  whose  abscissa  is  5  and 
ordinate  3,  making  an  angle  with  the  axis  of  abscissas  whose 
tangent  is  equal  to  2. 

Proposition  IV. — Theorem. 
(20.)   The  equation  of  a  straight  line  which  passes  through 
*     two  given  points  is 


y 


-Hy^-*1)- 


x'—x 


where  x'  and  y'  are  the  co-ordinates  of  one  of  the  given  points, 
x"  and  y"  the  co-ordinates  of  the  other  point,  and  x  and  y  the 
general  co-ordinates  of  the  line. 

Let  B  and  C  be  the  two  given  points, 
the  co-ordinates  of  B  being  x'  and  y',  and 
the  co-ordinates  of  C  being  x"  and  y". 
Then,  since  the  general  equation  for 
every  point  in  the  required  line  is 

y=ax+b,         (1) 
it  follows  that  when  the  variable  abscissa  x  becomes  x',  then 
y  will  become  y' ;  hence 

y'=ax'+b.         (2) 


On  the  Point  and   straight   Line.  29 

Also,  when  the  variable  abscissa  x  becomes  x",  then  y  be- 
comes y",  and  hence 

y"~aocf'+b.  (3) 

From  equations  (2)  and  (3)  we  may  obtain  the  values  of  a 
and  b,  and  substitute  them  in  the  first  equation.     Or  we  may 
accomplish  the  same  object  by  eliminating  a  and  b  from  the 
three  equations. 
If  we  subtract  equation  (2)  from  equation  (1),  we  obtain 

y-y'=a(x-x').  (4) 

Also,  if  we  subtract  equation  (3)  from  equation  (2),  we  obtain 
y'—y"=a(x'—x"), 

y'—y" 
from  which  we  find  a=  ,_  „. 

Substituting  this  value  of  a  in  equation  (4),  we  have 

v'  —  v" 

which  is  the  equation  of  the  line  passing  through  the  two  given 
points  B  and  C. 

y> y" 

(21.)  We  have  found  a  equal  to     ,_   ,,.     This  is  obvious 

from  the  figure.     For  y'—y"  is  equal  to  BD,  and  x'—x"  is  equal 

y'  —  y"  BD 

to  CD ;  hence  - — ^  is  equal  to  ^rp-,  which  is  the  tangent  of 
x'—x  ^D 

the  angle  BCD,  the  radius  being  unity  (Trigonometry,  Art.  42), 

If  the  origin  be  one  of  the  proposed  points,  then  x"=0,  and 

y"=0,  and  the  equation  becomes  . 

y' 

J     x' 
which  is  the  equation  of  a  straight  line  passing  through  the 
origin  and  through  a  given  point. 

Ex.  1.  Find  the  equation  to  the  straight  line  which  passes 
through  the  two  points  whose  co-ordinates  are  x'=l,  y'=4, 
x"  =  5,  y"=S,  and  determine  the  angle  which  it  makes  with  the 
axis  of  abscissas. 

Ex.  2.  Find  the  equation  to  the  straight  lino  which  passes 
through  the  two  points  x'— 2,  y'=3,  and  x"=4,  y"=b. 


30 


Analytical  Geometry. 


Proposition  V. — Theorem. 
(22.)   The  distance  between  two  given  points  is  equal  to 

V(x'-z"y+w-y»y, 

where  x'  and  y'  are  the  co-ordinates  of  one  of  the  given  points, 
and  x"  and  y"  those  of  the  other. 

Let  B  and  C  be  the  two  given  points,   y 
Designate  the  co-ordinates  of  B   by  x' 
and  y',  and  the  co-ordinates  of  C  by  x" 
and  y".     Draw  CD  parallel  to  AX.     The 
distance  BC  is  equal  to  A 

VCD2+BD2. 

But  CD=x'—x",  and  BD=y'—y"  ;  therefore  the  expression 
for  the  distance  between  B  and  C  is 


x 


V(x'-x"y  +  (y'-y")\ 

Proposition  VI. — Theorem. 

(23.)   The  tangent  of  the  angle  included  between  two  straight 
lines  is 


1+aa" 
where  a  and  a'  denote  the  tangents  of  the  angles  which  the 
Hvo  lines  make  with  the  axis  of  abscissas. 

Let  BC  and  DE  be  any  two  lines 
intersecting  each  other  in  P.  Let  the 
equation  of  the  line  DE  be 

y=ax+b, 
and  the  equation  of  BC  be 

y=a'x+b' ; 

then  a  will  be  the  tangent  of  angle  PEX,  and  a'  the  tangent 
of  the  angle  PCX.  Designate  the  angle  PEX  by  a,  and  the 
angle  PCX  by  a'.  Now,  because  PCX  is  the  exterior  angle  of 
the  triangle  PEC,  it  is  equal  to  the  sum  of  the  angles  CPE  and 
PEC  ;  that  is,  the  angle  EPC  is  equal  to  the  difference  of  the 
angles  PCX  and  PEX,  or 

EPC  =  PCX-PEX=a'-a  ; 

whence  tang.  EPC = tang.  (PCX- PEX) = tang,  (a' -a). 
But,  by  Trigonometry,  Art.  77, 


On    THE     iOINT    AND    STRAIGHT    LlNE.  Si 

tariff,  a'  — tangr.  a 


tang.  («'-«)= 5 5 — 

1+tang.  a  tang,  a.' 

Therefore,  tang.  EPC=^-^-. 

1+aa' 

If  the  angle  of  intersection  of  the  two  lines  be  a  right  angle, 
its  tangent  must  be  infinite.     But  in  order  that  the  expression 

-  may  become  infinite,  the  denominator  1+aa'  must  be- 


1+aa' 

come  zero;  so  that  in  this  case  we  must  have  aa'=  —  1,  or 

a= :.     This,  then,  is  the  condition  by  which  two  straight 

lines  are  shown  to  be  at  right  angles  to  each  other. 

(24.)  This  last  conclusion  might  have  been  derived  from  the 
principles  of  Trigonometry.  Thus,  let 
the  two  lines  PC,  PE  be  perpendicu- 
lar to  each  other ;  then  the  angle  PCE 
is  the  complement  of  PEC.  But  by 
(Trig.,  Art.  28)  tang.Xcotang.=R2  or 
unity;  hence  tang.  PEC X tang.  PCE 
=  1.  'Now  PCX,  being  the  supplement  of  PCE,  has  the  same 
tangent  (Trig.,  Art.  27),  but  with  a  negative  sign  (Trig.,  Art. 
70).     Hence 

tang.  PEC  X  tang.  PCX=-1. 

(25.)  The  equation  of  a  line  passing  through  a  given  point  is 

y— y'  =  a(x— x'). 
If  this  line  be  perpendicular  to  a  certain  given  line,  we  may 

for  a  substitute  — -„  where  a'  is  the  tangent  of  the  angle  which 

CI 

this  last  named,  line  makes  with  the  axis  of  abscissas.     Hence 

is  the  equation  ot  a  line  passing  through  a  given  point,  and  per    I 
pendicular  to  a  given  line. 

Proposition  VII. — Theorem.* 

(20.)   The  equation  of  a  straight  line  referred  to  ohlique  axes 
is 

y—ax+b. 


32  Analytical   Geometry. 

where  a  represents  the  ratio  of  the  sine  of  the  angle  which  the 
line  makes  with  the  axis  of  abscissas,  to  the  sine  oi  the  angle 
which  it  makes  with  the  axis  of  ordinates. 

Let  A  be  the  origin  of  co-ordinates, 
and  AX,  AY  oblique  axes,  and  PC  any 
straight  line  whose  equation  is  required 
to  be  determined.  Take  any  point  P  in 
the  given  line,  and  draw  PB  parallel  to 
AY ;  then  will  PB  be  the  ordinate,  and 
AB  the  abscissa  of  the  point  P.  From  E 
A  draw  AD  parallel  to  CP,  meeting  the  line  BP  in  D.  De- 
note the  angle  PEX,  or  its  equal  DAX,  by  a,  and  the  angle 
YAX  by  P. 

Since  PB  is  parallel  to  AY,  the  angle  ADB  is  equal  to  DAY ; 
that  is,  equal  to  (5— a. 

Let  AB=:r, 

BP=y, 

and  AC  or  DP=6. 

Then,  by  Trigonometry,  Theorem  I.,  Art.  49, 

BD  :  AB  :  :  sin.  a  :  sin.  (0— a), 

or  BD  .    x    ::  sin.  a  :  sin.  (j3— a). 

sin.  a 
Hence  BD=:r- — -p. r. 

sin.  (j3— a) 

But  BP=BD+DP. 

sin.  a 

Hence  ^=Xsm.(/3-a)+&- 

The  coefficient  of  x  in  this  equation  is  equal  to  the  sine  of 
I-he  angle  winch  the  line  makes  with  the  axis  of  X,  divided  by 
the  sine  of  the  angle  which  it  makes  with  the  axis  of  Y ;  and 
if  we  represent  this  factor  by  a,  the  equation  may  be  written 

y=ax+b, 
which  is  of  the  same  form  as  in  Theorem  L.  but  the  factor  a 
>  has  a  different  signification. 

ON  THE  TRANSFORMATION  OF  CO-ORDINATES. 

(27.)  When  a  line  is  represented  by  an  equation  in  reference 
to  any  system  of  axes,  we  can  always  transform  that  equation 
into  another  which  shall  equally  represent  the  line,  but  in  refer- 
ence to  a  new  system  of  axes  chosen  at  pleasure.     This  is 


z1 


/M' 


-X' 
-X 


On    the    Transform ation    of   Co-ordinates.    S',i 

called  the  transformation  of  co-ordinates  ;  and  may  consist 
either  in  altering  the  relative  position  of  the  axes  without 
changing  the  origin,  or  changing  the  origin  without  disturbing 
the  relative  position  of  the  axes  ;  or  we  may  change  both  the 
direction  of  the  axes  and  the  position  of  the  origin. 

Proposition  VIII. — Theorem. 
(28.)  The  formulas  f oi'  passing  from  one  system  of  co-ordinate 
axes  to  another  system,  respectively  parallel  to  the  first,  are, 

x=a+x', 

y=b+y', 

m  which  a  and  b  are  the  co-ordinates  of  the  new  origin. 

Let  AX,  AY  be  the  primitive  axes,  and         Y/      ,y' 
xet  A'X',  A'Y'  be  the  new  axes  to  which 
it  is  proposed  to  refer  the  same  line. 

Let  AB,  A'B,  the  co-ordinates  of  the 
new  origin,  be  represented  by  a  and  b : 
let  the  co-ordinates  of  any  point  P  relative  a  b  M 
to  the  primitive  axes  be  x  and  y,  and  the  co-ordinates  of  the 
same  point  referred  to  the  new  axes  be  x'  and  y'.  Then  we 
shall  have 

AM=AB+BM,  and  PM=MM'+PM'; 
that  is,  x=a+x',  and  y—b+y'i 

which  are  the  equations  required. 

The  new  origin  A'  may  be  placed  in  either  of  the  four  an- 
gles of  the  primitive  system,  by  attributing  proper  signs  to  a 
and  b. 

Proposition  IX. — Theorem. 
(29.)   The  f oi  mulcts  for  passing  from  a  system  of  rectangula 
co-ordinates  to  another  system  also  rectangular  are, 
x=x'  cos.  oi—y'  sin.  a, 
y—x'  sin.  a-f-y'  cos.  a, 
where  a  represents  the  angle  included 
between  the  two  axes  of  X. 

Let  AX,  AY  be  the  primitive  axes, 
and  AX',  AY'  be  the  new  axes,  and 
let  us  designate  the  co-ordinates  of  the 
point  P  referred  to  the  primitive  axes 

C 


34 


Analytical  Geometry. 


by  x  and  y,  and  its  co-ordinates  referred  to  the  new  axes  by 
x',  y .  Denote  the  angle  XAX'  by  a.  Through  P  draw  PR 
perpendicular  to  AX,  and  PR'  perpendicular  to  AX';  draw 
R'C  perpendicular,  and  R'B  parallel  to  AX. 

Then  AR=AC-CR. 

But  AR=x. 

Also,  AC  =  AR'Xcos.  XAX'=a;'  cos.  a, 

and  CR=BR'  =  PR'  sin.  BPR'=?/'  sin.  a. 

Hence  x=x'  cos.  a— y'  sin.  a. 

Also,  PR=BR  +  PB. 

But  PR=?/; 

BR=R'C  =  AR'  sin.  XAX'=.r'  sin.  «; 
and  PB  =  PR'  cos.  BPR'=y'  cos.  «. 

Hence  y~x'  sin.  <*+?/'  cos.  a. 

Scholium.  If  the  origin  be  changed  at  the  same  time  to  a 
point  whose  co-ordinates,  when  referred  to  the  primitive  sys 
tern,  are  a  and  b,  these  equations  will  become 

x=a+x'  cos.  a— y'  sin.  a, 

y=b+x'  sin.  a-fy'cos.  a. 

Proposition  X. — Theorem. 

(30.)  The  formulas  for  passing  from  a  system  of  rectangular, 
tc  a  system  of  oblique  co-ordinates,  are, 

x—x'  cos.  a+y'  cos.  a', 
y—x'  sin.  a+y'  sin.  a', 
where  a  and  a'  denote  the  inclination  of  the  new  axes  to  the 
primitive  axis  of  abscissas. 

Let  AX,  AY  be  the  primitive  axes,  y 
AX',  AY'  the  new  axes.  Denote  the 
angle  XAX'  by  a,  and  the  angle  XAY' 
by  a'.  Through  P  draw  PR  parallel 
to  AY,  and  PF  parallel  to  AY' ;  draw, 
also,  P'R'  parallel  to  AY,  and  P'B  par- 
allel to  AX. 

Then  AR  =AR'+R'R. 

But  AR  =x, 

AR'=AP'  cos.  XAX' 


and 
Hence 


-x  cos.  a, 
R'R=P'B=PP'  cos.  BP'P=y  cos.  a'. 

x=x'  cos.  a.-\-y'  COS.  a'. 


On   the    Transformation   of   Co-ordinates.    35 

Also,  PR=BR+PB. 

But  PR=y, 

BR=P'R'  =  AP'  sin.  XAX'=z'  sin.  a, 
and  PB=PP'  sin.  PP'B=y'  sin.  a'. 

Hence  .  y=x'  sin.  a+y'  sin.  a'. 

Scholium.  If  the  origin  be  changed  at  the  same  time  to  a 
point  whose  co-ordinates,  referred  to  the  primitive  system,  are 
a  and  b,  these  equations  will  become 

x=aJrx'  cos.  a+y'  cos.  a', 
y=b+x'  sin.  a+y'  sin.  a'. 

Proposition  XL — Theorem. 

(3 1 .)  The  formulas  for  passing  from  a  system  of  rectangular 
to  a  system  of  polar  co-ordinates  are, 

x=a+r  cos.  v, 
y=b+r  sin.  v, 

where  r  denotes  the  radius  vector,  and  v  the  angle  which  it 
makes  with  the  axis  of  abscissas. 

Let  AX,  AY  be  the  primitive  axes,  A'  y 
the  pole,  and  A'D,  parallel  to  AX,  be  the 
line  from  which  the  variable  angle  is  to  be 
estimated. 

Designate  the  angle  PA'D  by  v,  the  ra- 
dius vector  A'P  by  r,  the  co-ordinates  of   A      D  R 
the  point  P  referred  to  the  primitive  axes  by  x  and  y,  and  the 
co-ordinates  of  A'  by  a  and  b. 

Now  AR=AB+BR. 

But  BR= A'D= A'P  cos.  PA'D=r  cos.  v. 

Hence  x=a+r  cos.  v. 

Also,  PR=DR+PD. 

But  PD=A'P  sin.  PA'D=r  sin.  v. 

Hence  y=b+r  sin.  v. 

Scholium.  If  the  pole  A'  be  placed  at  the  origin  A,  these  equa- 
tions will  become 

x=r  cos.  v, 
y—r  sin.  v. 


SECTION   IV. 

ON   THE    CIRCLE. 

(32.)  A  circle  is  a  plane  figure  bounded  by  a  line,  every 
point  of  which  is  equally  distant  from  a  point  within  called  the 
center.  This  bounding  line  is  called  the  circumference  of  the 
circle.  A  radius  of  a  circle  is  a  straight  line  drawn  from  the 
center  to  the  circumference. 

Proposition  I. — Theorem. 

(33.)  The  equation  of  the  circle,  when  the  origin  of  co-ordi- 
nates is  at  the  center,  is 

a:»+ys=Ra ; 

where  R  is  the  radius  of  the  circle,  and  x  and  y  the  co-ordi- 
nates of  any  point  of  the  circumference. 

Let  A  be  the  center  of  the  circle  ;  it  is 
required  to  find  the  equation  of  a  curve 
such  that  every  point  of  it  shall  be  equal- 
ly distant  from  A.  Represent  this  dis- 
tance by  R,  and  let  x  and  y  represent 
the  co-ordinates  of  any  point  of  the 
curve,  as  P.  Then,  by  Geometry,  Prop. 
11,  B.  IV., 

AB2+BP2=AP2; 
that  is,  x*  +  y2  =R2, 

which  is  the  equation  required. 

(34.)  If  we  wish  to  determine  the  points  where  the  curve 
cuts  the  axis  of  X,  we  must  make 

y=0; 
for  this  is  the  property  of  all  points  situated  on  the  axis  of  ab- 
scissas.    On  this  supposition  we  have 

x=±R; 
which  shows  that  the  curve  cuts  the  axis  of  abscissas  in  two 
points  on  different  sides  of  the  origin,  and  at  a  distance  from  it 
equal  to  the  radius  of  the  circle. 


On    the   Circle.  37 

To  determine  the  points  where  the  curve  cuts  the  axis  of 
ordinates,  we  make  x=0,  and  we  obtain 

y=±R; 
which  shows  that  the  curve  cuts  the  axis  of  ordinates  in  two 
points  on  different  sides  of  the  origin,  and  at  a  distance  from  it 
equal  to  the  radius  of  the  circle. 

(35.)  If  we  wish  to  trace  the  curve  through  the  intermediate 
points,  we  reduce  the  equation  to  the  form 

y=±VR*-x\ 

Now,  since  every  value  of  x  furnishes  two  equal  values  of 
y,  with  contrary  signs,  it  follows  that  the  curve  is  symmetrical 
above  and  below  the  axis  of  X. 

If  we  suppose  x  to  be  positive,  the  values  of?/  continually  de 
crease  from  x—0,  which  gives  ?/=±R,  to  x=+K,  which  gives 
y  =  0. 

If  we  make  x  greater  than  R,  y  becomes  imaginary,  which 
shows  that  the  curve  does  not  extend  on  the  side  of  the  posi- 
tive abscissas  beyond  the  value  of  .r=+R. 

In  the  same  manner  it  may  be  shown  that  the  curve  does 
not  extend  on  the  side  of  the  negative  abscissas  beyond  the 
value  of.r=  —  R. 

Proposition  II. — Theorem. 

(36.)  The  equation  of  the  circle,  when  the  origin  is  on  the  cti- 
cumference,  and  the  axis  of  x  passes  through  the  center,  is 

*f=2Rx-x\ 
where  R  is  the  radius  of  the  circle,  and  x  and  y  the  co-ordi- 
nates of  any  point  of  the  circumference. 

Let  the  origin  of  co-ordinates  be  at  A,  y 
a  point  on  the  circumference  of  the  cir- 
cle. Draw  AX,  the  axis  of  abscissas, 
through  the  center  of  the  circle.  Let 
P  be  any  point  on  the  circumference, 
and  draw   PB   perpendicular  to   AX.       V  J 

Denote  the  line  AD  by  2R,  the  distance  ^ S 

AB  by  x,  and  the  perpendicular  BP  by  y ;  then  BD  will  be 
represented  by  2R— x. 

Now  BP  is  a  mean  proportional  between  the  segments  AB 
and  BD  (Geom.,  Prop.  22,  Cor.,  B.  IV.) ;  that  is. 


3S 


Analytical   Geometry. 


BP=ABxBD, 
or  y*=x(2R-x)=2Rx-x\ 

which  is  the  equation  required. 

(37.)  If  we  wish  to  determine  where  the  curve  cut,*  the  axis 
of  X,  we  make  y—0,  and  we  obtain 
z(2R— x)  =  0. 

This  equation  is  satisfied  by  supposing  x=0,  or  2R— x=0, 
from  the  last  of  which  equations  we  derive  r=2R.  The  curve, 
therefore,  cuts  the  axis  of  abscissas  in  two  points,  one  at  the 
origin,  and  the  other  at  a  distance  from  it  equal  to  2R. 

To  determine  where  the  curve  cuts  the  axis  ofordinates,  we 
make  x=0,  which  gives 

y=o, 

which  shows  that  the  curve  meets  the  axis  of  urdinates  in  but 
one  point,  viz.,  the  origin. 


Proposition  III. — Theorem. 

(38.)    The  most  general  equation  of  the  circh  is 

{x-xy+{y-yy=R\ 

where  R  denotes  the  radius  of  the  circle,  x'  and  y'  are  the  co- 
ordinates of  the  center,  and  x  and  y  the  co-ordinates  of  any 
point  of  the  circumference. 

Let  C  be  the  center  of  the  circle,  and  Y 
assume  any  rectangular  axes  AX,  AY. 
Let  the  co-ordinates  AB,  BC  of  the  cen- 
ter be  denoted  by  x'  and  y' ;  while  the 
co-ordinates  of  any  point  P  in  the  cir- 
cumference are  denoted  by  x  and  y. 
Then,  if  we  draw  the  radius  CP,  and   Ar~  jj       '       X 

CD  parallel  to  the  axis  of  X,  we  shall  have 

CD^=x—x', 
and  VD—y—ij'. 

But  CDa+PDs=CP\ 

Hence  we  have  (x— x'y-\-{y— y'Y—R2, 

which  is  the  equation  sought. 

(39.)  To  find  the  points  where  the  curve  intersects  the  axis 
of  X,  we  must  make  y=0,  which  gives, 
(x-x'Y+y'^K2, 


0\   the   Circle. 


30 


whence 


(x-x'Y=R*-y 


x=x' 


or  j;=z'±VR"-y'1, 

where  we  see  that  the  values  of  £  will  become  imaginary  when 
y'  exceeds  R ;  and  it  is  evident  that  if  the  distance  of  the  cen- 
ter of  the  circle  from  the  axis  of  abscissas  exceeds  the  radius 
of  the  circle  there  can  be  no  intersection. 

To  find  the  point  where  the  curve  intersects  the  axis  of  Y, 
we  must  make  x=0,  which  gives 

y=y'±  VR'-x'"; 
which  becomes  imaginary  when  x'  exceeds  R,  and  it  is  plain 
that  in  this  case  there  can  be  no  intersection. 


Proposition  IV. — Theorem. 
(40.)    The  equation  of  a  tangent  line  to  the  circle  is 
xx'-\-yy'=JV, 

where  R  denotes  the  radius  of  the  circle,  x'  and  y'  are  the  co- 
ordinates of  the  point  of  contact,  and  x  and  y  are  the  general 
co-ordinates  of  the  tangent  line. 

Let  BC  be  a  line  touching  the 
circle,  whose  center  is  A,  in  the 
point  P.  Let  the  co-ordinates  of 
the  point  P  be  x'  and  y',  and  draw 
the  radius  AP.  The  equation  of 
the  line  AP,  passing  through  the 
origin  and  through  the  point  x',  y', 
Art.  21,  is 

y' 

y=x<x- 

Now  a  tangent  is  perpendicular  to  the  radius  at  the  point  of 
contact  (Geom.,  Prop.  IX.,  B.  III.).  But  the  equation  of  a 
line  passing  through  a  given  point,  and  perpendicular  to  a 
given  line,  Art.  25,  is 

y-y'=--(x-x'). 

The  value  of  a',  taken  from  the  equation  of  the  radius,  is 

y' 


40 


Analytical  Geometry. 


1         x> 

cience  := :. 

a'         y1 

The  equation  of  the  tangent  line  is,  therefore, 

x' 

Clearing  of  fractions  and  transposing,  we  obtain 

xx'  -\-yy'= x'2 + yn. 
But  since  the  point  P  is  on  the  circumference,  its  co-ordinates 
must  satisfy  the  equation  of  the  circle  ;  that  is, 

x'2  +  ij'"-=R\ 
Hence  xx'+yy'=~R."\ 

which  is  the  equation  required. 

(41.)  The  equation  of  the  tangent  may  also  be  obtained, 
without  employing  the  geometrical  property  above  referred  to, 
by  a  method  which  is  applicable  to  all  curves  whatever. 

Let  us  first  consider  a  line  BC, 
meeting. the  curve  in  two  points  P' 
and  P;/ ;  the  co-ordinates  of  P'  be- 
ing represented  by  x',  y',  and  those 
of  P"  by  x",  y".  The  equation  of 
the  line  BC,  Art.  20,  is 

y-2/-fr^-<>;      (1) 

and,  since  both  the  points  P'  and  P" 
are  on  the  circumference,  we  must  have 

xn  +y"  =R\        *  (2) 

and  x"*+ij"'=R\  (3) 

Subtracting  equation  (3)  from  equation  (2),  we  obtain 
ijn-y"2+xr2-x"*=0  ; 

that  is,        (y'+y")  (y'-y")+(x'+x")  (x'-x")=o. 

y'—y"        x'+x" 

whence  — -— — — -. 

x —x         y +y 

Substituting  this  value  in  equation  (1),  we  obtain 


y-y- 


x'+x",  _   ,, 

'y'+y"{X    Xh 


(4) 


If  dow  we  suppose  the  secant  BC  to  move  toward  the  point 
P,  tne  point  P'  will  approach  P" ;  and  when  P'  coincides  with 
P",  the  secant  line  will  become  a  tanoent  to  the  circumference. 


On    THE    C I B  C L E. 


41 


When  this  takes  place,  x'  will  equal  x",  and  y'  will  equal  y'\ 
and  the  last  equation  becomes 

x' 

y-y'=--,(x-*'), 

as  before  found. 

(42.)  To  determine  the  point  in  which  the  tangent  intersects 
the  axis  of  X,  we  make  y=0,  which  gives 

xx'—TL"', 
R2 


or 


x=— =AC. 

x' 


To  determine  the  point  in  which 
the  tangent  intersects  the  axis  of  Y, 
we  make  x=0,  which  gives 

yy'=R>, 


or 


R2 

y=— =AB. 

y 


Proposition  V. — Problem. 

(43.;   Given  the  base  of  a  triangle,  and  the  sum  of  the  squares 
of  its  sides,  to  determine  the  triangle. 

Let  AB  be  the  base  of  the  proposed  trian- 
gle. Bisect  AB  in  C  ;  draw  CY  perpendicu- 
.ar  to  AB,  and  assume  YC,  CB  as  a  system 
of  rectangular  axes. 

Let  x  and  y  be  the  co-ordinates  of  P,  the 
vertex  of  the  triangle,  and  from  P  let  fall  the 
perpendicular  PD.  Let  a  denote  AC  or  CB, 
and  put  m  for  the  sum  of  the  squares  of  the 
sides  AP,  BP. 

Then,  by  Geom.,  Prop.  XL,  B.  IV.,  we  shall  have 

PD2+AD2=AP2, 
and  PD2+BD2=BP2; 

or  y-  +  (x+ay  =  AV\ 

and  if  +  (x-ay=B¥\ 

Adding  these  equations  together,  we  obtain 

2y1-\-2x*+2a'=AVi  +  BV'=m. 


42 


Analytical  Geometry. 


Whence 


.    m 


Comparing  this  result  with  Art.  33,  we  see  that  this  equation 
represents  a  circle  whose  center  is  the  origin  C,  and  the  radius 


V- 


so  that  if  this  circle  be  described,  and  lines  be  drawn  from  A 
and  B  to  any  point  in  its  circumference,  a  triangle  will  be 
formed  which  satisfies  the  proposed  conditions. 

Proposition  VI. — Theorem. 

(44.)    The  polar  equation  of  the  circle,  when  the  origin  is  on 
the  circumference,  is 

r=2R  cos.  v, 
where  R  represents  the  radius"of  the  circle,  r  the  radius  vector, 
and  v  the  variable  angle. 

The  equation  of  the  circle  referred  to  rectangular  axes,  when 
the  origin  is  on  the  circumference,  Art.  36,  is 

y'=2Rx-x\  (1) 

Let  A  be  the  position  of  the  pole, 
and  AX  the  line  from  which  the  varia- 
ble angle  is  estimated.  The  formulas 
for  passing  from  a  system  of  rectan- 
gular to  a  system  of  polar  co-ordi- 
nates, the  origin  remaining  the  same, 
Art.  31,  are 

•  x=r  cos.  v, 

y=r  sin.  v. 
Squaring  each  member  of  these  equations,  and  substituting 
the  values  of  a;3,  ?/2,  thus  found  in  equation  (1),  we  obtain 

r~  sin.2  u=2Rr  cos.  v— r2  cos.2  v ; 
or,  by  transposition,  r2(sin.2  u+cos.2  v)—2Rr  cos.  v. 

But  sin.2  u  +  cos.2  v  is  equal  to  unity. 
Hence  r2=2Rr  cos.  v  ; 

or,  dividing  by  r,  we  obtain 

r=2R  cos.  v, 
which  is  the  polar  equation  of  the  circle. 

(45.)   This  equation  might  have  been  derived  directly  from 
the  figure.     Thus,  by  Trig.,  Art.  41, 


On   the    C  ni c  L  E.  43 

radius  :  AB  : :  cos.  BAP  :  AP, 
or  1  :  2R  : :  cos.      v     :  r ; 

whence  r=2R  cos.  v. 

(40.)   When  v=0,  the  cos.  v=l,  and  we  have 
r=2R=AB. 
As  v  increases  from  0  to  90°,  the  radius  vector  determines  all 
the  points  in  the  semicircumference  BPA ;  and  when  u=90°, 
then  cos.  v=0,  and  we  have 

r=0. 
From  u=270°  to  v=360°,  the  radius  vector  will  determine 
all  the  points  of  the  semicircumference  below  the  axis  of  ab- 
scissas. 

Examples. 

1.  On  a  circle  whose  radius  is  6  inches,  a  tangent  line  is  drawn 
through  the  point  whose  ordinate  is  4  inches  :  see  figure,  Art. 
42.     Determine  where  the  tangent  line  meets  the  two  axes. 

Ans.  AC=  ;  AB  = 

2.  Find  the  angle  which  the  tangent  line  in  the  preceding  ex- 
ample makes  with  the  axis  of  X. 

Ans. 

3.  Find  the  point  on  the  circumference  of  a  circle  whose  ra- 
dius is  5  inches,  from  which,  if  a  radius  and  a  tangent  line  be 
drawn,  they  will  form,  with  the  axis  of  X,  a  triangle  whose  area 
is  35  inches. 

Ans.  Abscissa  =  ;  ordinate  = 

4.  The  radius  of  a  circle  is  5  inches,  and  the  variable  angle 
is  36  degrees,  the  pole  being  on  the  circumference  ;  determine 
the  radius  vector. 

Ans. 

5.  The  radius  of  a  circle  is  5  inches,  and  the  radius  vector  is 
8  inches  ;  determine  the  variable  angle. 

Ans. 

6.  The  radius  vector  of  a  circle  is  16  inches,  and  the  variable 
angle  is  42  degrees  ;  determine  the  radius  of  the  circle. 

Ans.  , 


SECTION    V. 


ON   THE   PARABOLA. 

(47.)  A  parabola  is  a  plane  curve,  every  point  of  which  is 
equally  distant  from  a  fixed  point  and  a  given  straight  line. 

The  fixed  point  is  called  the  focus  of  the  parabola,  and  the 
given  straight  line  is  called  the  directrix. 

Thus,  if  F  be  a  fixed  point,  and  BC  a  b 
given  line,  and  the  point  P  move  about  F  D 
in  such  a  manner  that  its  distance  from  F  is 
always  equal  to  the  perpendicular  distance 
from  BC,  the  point  P  will  describe  a  parab- 
ola of  which  F  is  the  focus  and  BC  the  di- 
rectrix. 

The  distance  of  any  point  of  the  curve 
from  the  focus,  is  called  the  radius  vector 
of  that  point. 

(48.)  From  the  definition  of  the  parabola,  the  curve  may  be 
described  mechanically. 

Let  BC  be  a  ruler  laid  upon  a  plane,  and 
let  DEG  be  a  square.  Take  a  thread 
equal  in  length  to  DG,  and  attach  one  ex-  D 
tremity  at  G,  and  the  other  at  some  point, 
as  F.  Then  slide  the  side  of  the  square 
DE  along  the  ruler  BC,  and  at  the  same 
time  keep  the  thread  continually  tight  by 
means  of  the  pencil  P;  the  pencil  will  de- 
scribe one  part  of  a  parabola,  of  which  F 
is  the  focus,  and  BC  the  directrix.  For  in  every  position  of 
the  square, 

PF+PG=PD+PG, 
and  hence  PF=PD; 

that  is,  the  point  P  is  equally  distant  from  the  focus  F  and  the 
directrix  BC. 

If  the  square  be  turned  over  and  moved  on  the  other  side  of 


On   the    Parabola. 


15 


:; 


;- 


the  point  F,  the  other  part  of  the  same  parabola  may  be  de 
scribed. 

(49.)  A  diameter  is  a  straight  line  drawn  through  any  point 
of  the  curve  perpendicular  to  the  directrix.  The  vertex  of  the 
diameter  is  the  point  in  which  it  cuts  the  curve. 

The  axis  of  the  parabola  is  the  diameter  which  passes 
through  the  focus. 

v  The  parameter  of  a  diameter  is  the  double  ordinate  which 
passes  through  the  focus. 


Proposition  I. — Theorem. 

(50.)    The  equation  of  the  parabola,  referred  to  rectangular 
axes  whose  origin  is  at  the  vertex  of  the  axis,  is 

y*=2px, 
where  x  and  y  are  the  general  co-ordinates  of  the  curve,  and 
2p  is  the  parameter  of  the  axis. 

Let  F  be  the  focus,  and  DC  the  di- 
rectrix. Take  AX  as  the  axis  of  ab- 
scissas, and  let  the  origin  be  placed  at  A, 
the  middle  point  of  BF.     Represent  BF 

by  p,  whence  AF  will  equal  — .     Let  x 

and  y  be  the  co-ordinates  of  any  point  P 
in  the  curve,  and  represent  FP  by  r. 
By  the  definition  of  the  curve, 


Also, 
But 
that  is, 


PF=PD=AR+AB=x+|. 

FR=s-f. 

PR2+FR2=PF2; 


y'+ 


(*-!)'=(-: 


Whence,  by  expanding,  we  obtain 

y*=2px. 
(51.)   Cor.  I.  If  we  make  x=0,  we  have 

y=0, 
which  shows  that  the  curve  passes  through  the  origin  A. 

P 

If  we  make  x=jr,  we  shall  have 


46 


Analytical    Geometry. 


y-p< 

or  y—  p ; 

whence  2y=2p ; 

that  is,  the  constant  quantity  2p,  called  the  parameter,  is  equal 

to  the  double  ordinate  through  the  focus,  conformably  to  the 

definition,  Art.  49. 

Cor.  2.  From  the  equation  of  the  parabola  we  obtain 
y=±  V2px, 
which  shows  that  for  every  value  of  x  there  will  be  two  equal 
values  of  y,  with  contrary  signs.     Hence  the  curve  is  sym- 
metrical with  respect  to  the  axis  of  X. 

Cor.  3.  If  we  convert  the  equation  y2=2px  into  a  propor 
lion,  we  shall  have 

x  :  y  : :  y  :  2p  ; 
that  is,  the  parameter  of  the  axis  is  a  third  proportional  to  any 
abscissa  and  its  corresponding  ordinate. 

Cor.  4.   The  squares  of  ordinates  to  the  axis  are  to  each  other 
as  their  corresponding  abscissas. 

Designate  any  two  ordinates  by  y'.  y",  and  the  correspond 
ng  abscissas  by  x',  x",  then  we  shall  have 

y12  =2px', 
and  y"2=2px". 

Hence  y'2  :  y"*  : :  2px'  :  2px"  : :  x'  :  x". 

Proposition  II. — Theorem. 

(52.)   The  equation  of  a  tangent  line  to  the  parabola  is 
yy'=p{x+x'). 
where  x',  y'  are  the  co-ordinates  of  the  point  of  contact,  and  p 
is  half  the  parameter  of  the  axis. 

Draw  any  line  P'P",  cutting  the 
parabola  in  the  points  P',  P"  ;  if  this 
line  be  moved  toward  P,  it  will  ap- 
proach the  position  of  the  tangent, 
and  the  secant  will  become  a  tangent 
when  the  points  P',  P"  coincide. 

Let  x',  y'  be  the  co-ordinates  of  the 
point  P',  and  x",  y"  the  co-ordinates 
of  the  point  P".     The  equation  of  the  line  passing  through 
these  two  points,  Art.  20,  will  be 


On    the    Pauabola.  4T 

,     v' —  y" 

This  is  the  general  equation  of  a  straight  line  passing  through 
two  given  points,  and  has  no  special  reference  to  the  parabola. 
fn  order  to  make  it  the  equation  of  a  secant  line  to  the  parab- 
ola, we  must  deduce  from  the  equation  of  the  curve  the  value 
of  the  coefficient  of  x—x',  and  substitute  it  in  equation  (1). 
Thus,  since  the  points  P'  and  P"  are  on  the  curve,  we  shall 
have 

y"  =2px'  (2). 

y"*=2px"  (3). 

Subtracting  equation  (3)  from  (2),  we  nave 
yl2-y""-=2p(x'-x"). 

Whence  £z^=_^_, 

x'—x       y' ' -\-y" 

Substituting  this  value  in  equation  (1),  the  equation  of  the 

secant  line  becomes 

2v 

y-y'=y-r^-^~x')-       (4) 

The  secant  will  become  a  tangent  when  the  points  P',  P" 
coincide,  in  which  case 

x'=x"  and  y'—y". 
Equation  (4)  in  this  case  becomes 

which  is  the  equation  of  a  tangent  to  the  parabola  at  the  point 
P.     If  we  clear  this  equation  of  fractions,  we  have 

yy'-y"=px-px'. 

But  y"=2px'. 

Hence  yy'=px—  px'+2px', 

or  yy'=p{x-\-x'). 

(53.)  Definition.  A  subtangent  is  that  part  of  a  diameter 
intercepted  between  a  tangent  and  ordinate  to  the  point  of 
contact. 

Cor.  1.  To  find  the  point  in  which  the  tangent  intersects  the 
axis  of  abscissas,  make  y=0  in  the  equation  of  the  tangent, 
and  we  have 

0=p(x+x') 


48 


Analytical   Geometry. 


that  is,  x=—x'. 

or  AT=-AR; 

that  is,  the  subtangent  is  bisected  at  the  vertex. 

Cor.  2.  This  property  enables  us  to 
diaw  a  tangent  to  the  curve  through  a 
given  point.  Let  P  be  the  given  point; 
from  P  draw  PR  perpendicular  to  the 
axis,  and  make  AT=AR.  Draw  a  line 
through  P  and  T,  and  it  will  be  a  tan- 
gent to  the  parabola  at  P. 

Schol.  In  the  equation 

P 

—  represents  the  trigonometrical  tangent  of  the  angle  which 

the  tangent  line  makes  with  the  axis  of  the  parabola. 

(54.)  Definitions.  A  normal  is  a  line  drawn  perpendicular 
to  a  tangent  from  the  point  of  contact,  and  terminated  by  the 
axis. 

A  subnormal  is  the  part  of  the  axis  intercepted  between  the 
normal  and  the  corresponding  ordinate. 


Proposition  III. — Theorem. 
(55.)   The  equation  of  a  normal  line  to  the  parabola  is 

y' 
y-y'=—-(x-x'), 

where  x',  y'  are  the  co-ordinates  of  the  point  of  intersection 
with  the  curve. 

The  equation  of  a  straight  line 
passing  through  the  point  whose  co- 
ordinates are  x',  y',  Art.  18,  is 

y-y'  =  a(x-x')',  (1) 

and,  since  the  normal  line  is  per-   T- 
pendicular  to  the  tangent,  we  shall 
have,  Art.  23, 

1 

a= -. 

a 

But  we  have  found  for  the  tangent  line,  Prop   II.,  Schol. 


On   the   Parabola. 


49 


Hence 


a 

P 

y' 

a— 

y' 

V 


Substituting  this  value  in  equation  (1),  we  shall  have  for  the 
^quation  of  the  normal  line 


y 
y-y'=--(x-x'). 


(2) 


(56.)  Cor.  To  find  the  point  in  which  the  normal  intersects 
the  axis  of  abscissas,  make  y=0  in  equation  (2),  and  we  have, 
aftei  reduction, 

x— x'=p. 

But  x  is  equal  to  the  distance  AN,  and  x'  to  AR ;  hence 
x—x'=p  is  equal  to  RN  ;  that  is,  the  subnormal  is  constant,  and 
equal  to  half  the  parameter  of  the  axis. 

Proposition  IV. — Theorem. 

(57.)  The  normal,  at  any  point  of  the  parabola,  bisects  the  an 
gle  made  by  the  radius  vector  and  the  diameter  passing  through 
that  point. 

Let  PT  be  a  tangent  to  a  parabola, 
PF  the  radius  vector,  PN  the  normal, 
and  PB  the  diameter  to  the  point  P ; 
the  normal  PN  bisects  the  angle  BPF. 
Let  x'  represent  the  abscissa  of  the 
point  P. 

Now     FN-AR+RN-AF. 

But    KK=x',  RN=p,  and  AF=-|. 


Hence 
But  in  Prop.  I.  we  found 


FN=x'+p-j-=x'+±. 


Hence  FN=FP. 

Therefore  tne  angle  FPN=FNP=the  alternate  angle  BPN, 

(W.)   Cor.  FR=AR-AF=rr'-|. 


But  ril=2x',  Prop.  II.,  Cor.  1. 


D 


so 


Analytical  Geometry. 


Hence 


TF=TR-FR= 


:s'+f=PF; 


that  is,  if  a  tangent  to  the  parabola  cut  the  axis  produced,  the 
points  of  contact  and  of  intersection  are  equally  distant  from 
the  focus. 

Proposition  V. — Theorem. 

(59.)  If  a  perpendicular  be  drawn  from  the  focus  to  any  tan- 
gent, the  perpendicular  will  be  a  mean  proportional  between  the 
distances  of  the  focus  from  the  vertex  and  from  the  point  of 
contact. 

Let   FB  be  a   perpendicular   drawn 
.from  the  focus  to  the  tangent  PT.     Join 
AB,  and  draw  the  ordinate  PR. 

Since  FT  is  equal  to  FP  (Prop.  IV., 
Cor.),  and  FB  is  drawn  perpendicular  ^ 
to  PT,  PB  is  equal  to  BT.     But  RA  is 
equal  to  AT,  Prop.  II.,  Cor.  1  ;  hence 

TB  :  BP  : :  TA  :  AR, 
and  therefore  AB  is  parallel  to  PR.  But  PR  is  perpendicular 
to  the  axis  ;  hence  AB  is  perpendicular  to  TF ;  and  therefore, 
by  similar  triangles,  FAB,  FBT,  we  have 

FA  :  FB  : :  FB  :  FT  or  FP. 

Proposition  VI. — Theorem. 

(60.)  The  equation  of  the  parabola  referred  to  a  tangent  line, 
and  the  diameter  passing  through  the  point  of  contact,  the  origin 
being  the  point  of  contact,  is 

y'=2p'x, 

where  2p'  is  the  parameter  of  the  diameter  passing  through 
the  origin.  • 

The  formulas  for  passing  from  rectangular  to  oblique  axes 
are  (Art.  30,  Schol), 

x=a+x'  cos.  a+y'  cos.  a',  (1) 

y=b+x'  sin.  a-\-y'  sin.  a1.  (2) 

Since  the  new  origin  is  to  be  on  the  curve,  its  co-ordinates 
must  satisfy  the  equation  of  the  curve  ;  that  is, 

li=2pa,  whence  a— rr. 


On   the   Parabola.  51 

Also,  <since  every  diameter  is  parallel  to  the  axis,  we  must 
have 

a=0; 
whence  sin.  a=0,  and  cos.  a=l. 

And,  since  the  tangent  of  the  angle  which  a  tangent  line  makes 

with  the  axis  of  the  parabola  (Prop.  II.,  Schol.)  is  — ,  we  must 

have 

p        p  sin.  a' 

~  or  4=tan£r.  a'-— 


y'        b  cos.  a' 

b  sin.  a1 
whence  cos.  a'=— 


P 

Making  these  substitutions,  formulas 
(1)  and  (2)  become 

&2  by'  sin.  a! 

2p  p 

and  y=bJry'  sin.  a'. 

Substituting  these  values  in  the  gen- 
eral equation  of  the  parabola 

y*=2px, 
we  have 

V+2by'  sin.  a'+y"  sin.2  a' =b2 +2px' +2by'  sin.  a' ; 
or  ?/'2  sin.2  a'=2px' ; 

whence  3//2=-^i — j- 


If  we  put  p'—- — r— ,  and  omit  the  accents  of  the  variables, 

r      r      sin.1  aJ 

we  shall  have 

y2=2p'x, 
which  is  the  equation  required ;  where  2p'  is  called  the  pa- 
rameter of  the  diameter  A'X'.     See  Art.  63. 

(61.)   Cor.   The  squares  of  ordinates  to  any  diameter  are  to 
each  other  as  their  corresponding  abscissas. 

Designate  any  two  ordinates  by  y',  y",  and  the  correspond- 
ing  abscissas  by  x',  x",  we  shall  have 

yn  =2p'x', 
and  y"2=2p'x". 

Hence  yn  ;  y"2  ■  •  2p'x'  :  2p'x"  : :  x'  :  x". 


52 


Analytical  Cjeometiv  y. 


Proposition  VII. — Theorem. 
(62.)   The  parameter  of  any  diameter  is  equal  to  four  tunes 
the  distance  from  the  vertex  of  that  diameter  to  the  focus. 
We  have  from  the  last  Proposition, 
p     sin.  a! 
b     cos.  a'' 
whence      b  sin.  a'=p  cos.  a', 
and  ¥  sin.8  a'=p2  cos.2  a', 

=j»a(l  —  sin.2  a'), 
=p*—p*  sin.2  a'. 

Therefore     sin.sa'=— - — ;. 

But  b1—2ap,  from  the  equation  of  the  curve. 

Hence  sin.2  a'=     r       —    r    . 

2ap+p*     2a+p 

Now  »'  was  taken  equal  to  -r-^ — :  (Art.  60). 
7  '  sin.2  a'  v  ' 

Hence  p'—2a-\-p,  » 

and  2P'=4(«+f)- 

But  a  +  ^  is  equal  to  A'F  (Prop.  I.). 

Hence  2p',  or  the  parameter  of  the  diameter  A'X',  is  equa. 
to  4A'F. 

(63.)  Scholium.  If  through  the  focus 
F  the  line  BD  be  drawn  parallel  to  the 
tangent  TA',  then  calling  x  and  y  the 
co-ordinates  of  the  point  D, 

.r=A'C  =  TF=A'F  (Prop.  IV.,  Cor.), 

=|  (Prop.  VII.). 
But,  by  Prop.  VI.,  y"=2p'x. 
Hence  ..»_«-./ w  J1 


y 


=2p'X^=p'% 


or  y~p', 

and  2y~2p'; 

that  is,  the  quantity  2p'.  which  has  been  called  the  parameter 

of  the  diameter  A'X',  is  equal  to  the  double  ordinate  passing 

through  the  focus,  conformably  to  the  definition,  Art.  49. 


On   the   Parabola. 


53 


Proposition  VIII. — Theorem. 
(64.)   The  polar  equation  of  the  ■parabola,  the  pole  being  at 
the  focus,  is   ' 

r= P , 

l-*-cos.  v' 

xvnerep  represents  half  the  parameter,  and  v  is  the  angle  which 
the  radius  vector  makes  with  the  axis. 

We  have  found  the  distance  of  any  point 
of  the  parabola  from  the  focus,  Prop.  I.,  to  be 

r=FP=a:+|, 

where  the  abscissa  x  is  reckoned  from  the 
vertex  A.     In  order  to  transfer  the  origin 

from  A  to  F,  we  must  substitute  for  x,  x'+— ; 

whence  r=x'+p. 

If  we  represent  the  angle  PFA'  by  v,  we  shall  have  (Trig., 
Art.  41) 

xr=-j-r  cos.  v ; 
whence  r=p-i~r  cos.  v, 

V 


or 


r=: 


1 — cos.  v 
^  'wfeem.the  angle  v  is  estimated  from  the  vertex  A  toward  the 

Proposition  IX. — Theorem. 

(G5.)   The  area  of  any  segment  of  a  parabola  is  equal  to  two 
thirds  of  the  rectangle  described  on  its  abscissa  and  ordinate. 

Let  APR  be  a  segment  of  a  parab-    M 
ola  bounded  by  the  axis  AR  and  the 
ordinate  PR.      Complete  the  rect-  ^ 
angle  AMPR ;  then  will  the  parab-   m'" 
olic  segment  APR  be  two  thirds  of 
the  rectangle  AMPR. 

Inscribe  in  the  parabola  a  polygon 
PP'P". . .  AR,  and  through  the  points 
P,  P',  P",  etc.,  draw  parallels  to  AR 
and  PR,  forming  the  interior  rectan- 


54  Analytical   Geometry. 

gfes  PR,  P  'R',  etc.,  and  the  corresponding  exterior  rectangles 
P  M,  P"M',  etc.  Designate  the  former  by  P,  P',  P",  etc. ;  the 
latter  by  p,  p',  p",  etc.,  and  the  corresponding  co-ordinates  by 
x,  y,  x',  y't  etc.,  we  shall  then  have 

P'R=P'R'xR'R, 

or  V=y'(x— x') 

Also,  P'M=PM'XMM', 

or  P=x'(y-y') ; 

.  P    y'(x—x') 

which  gives  -=^77 K-  0) 

p    x'{y-y') 

But,  since  the  points  P,  P',  etc.,  are  on  the  curve,  we  have 

y*=2px,  y'2=2px' ; 

y1— y»  y'* 

whence  x— x'=— - — ,  and  x'=—. 

2p  2p 

Substituting  these  values  in  equation  (1),  we  obtain 

p  y'(y*-y'l   y+y'   ,  ,  y 


p   y,2(y-y')     y'         y 

In  the  same  manner  we  find 

P'  y< 

v        y 

P"  y" 

— =1+-^.  etc. 
p"  y" 

,If,  now,  we  suppose  the  vertices  of  the  polygons  P,  P',  P' 
etc.,  to  be  so  placed  that  the  ordinates  shall  be  in  geometrical 
progression,  we  shall  have 

y__y^_r_   t 
y    y    y 

so  that  each  interior  rectangle  has  to  its  corresponding  exterior 

y 

rectangle  the  ratio  of  \-\ — -  to  1. 

y 

Therefore,  by  composition, 

P+P^-P"+,etc.,r_l     y 
p+p'+p"+,  etc.,  3/'' 

that  is,  the  sum  of  all  the  interior  rectangles  is  to  the  sum  of 

y 

all  the  exterior  rectangles  as  H — :  to  1. 

a  y, 

The  nearer  the  points  P,  P',  P"  are  taken  to  each  other,  the 
nearer  does  the  sum  of  the  interior  rectangles  approach  to  th'j 


On   the   Parabola.  £5 

v 
area  of  the  parabolic  segment,  and  the  ratio  --  approaches  to 

a  ratio  of  equality.     Hence,  designating  the  area  APR  by  S, 
and  the  area  AMP  by  s,  we  have 


or 


s_ 

1  +  1=2. 

s 

s 
-=3; 

s= 

|(S+#). 

whence 

But  S-fs  is  equal  to  the.  area  of  the  rectangle  AMPR  ?  hence 
the  parabolic  segment  is  two  thirds  of  the  area  of  the  circum- 
scribing rectangle. 

Examples. 

1.  On  a  parabola,  the  parameter  of  whose  axis  is  10  inches,  a 
tangent  line  is  drawn  through  the  point  whose  ordinate  is  6  inch- 
es, the  origin  being  at  the  vertex  of  the  axis  ;  determine  where 
the  tangent  line  meets  the  two  axes  of  reference. 

2.  Determine  wh,ere  the  normal  line,  passing  through  the  same 
point  as  in  the  preceding  example,  will,  if  produced,  meet  the 
two  axes. 

3.  Find  the  angle  which  the  tangent  line  in  the  first  example 
makes  with  the  axis  of  X. 

4.  Find  the  point  on  the  curve  of  a  parabola  whose  param- 
eter is  10  inches,  from  which,  if  a  tangent  and  normal  be  drawn, 
they  will  form  with  the  axis  of  X  a  triangle  whose  area  is  36 
inches. 

5.  On  a  parabola  whose  parameter  is  10  inches,  find  the  point 
from  which  a  tangent  line  must  be  drawn  in  order  that  it  may 
make  an  angle  of  35  degrees  with  the  axis  of  the  parabola. 

6.  The  parameter  of  a  parabola  is  10  inches,  and  the  variable 
angle  is  144  degrees,  the  pole  being  at  the  focus  ;  determine  the 
radius  vector. 

7.  The  parameter  of  a  parabola  is  10  inches,  and  the  radius 
vector  is  25  inches  ;  determine  the  variable  angle. 

8.  The  radius  vector  of  a  parabola  is  25  inches,  and  the  vari- 
able angle  is  144  degrees;  determine  the  parameter  of  the  pa- 
rabola. 


m 


SECTION   VI. 

On  the  ellipse. 

(66.)  An  ellipse  is  a  plane  curve  in  which  the  sum  of  the 
distances  of  each  point  from  two  fixed  points  is  equal  to  a 
given  line.     The  two  fixed  points  are  called  the  foci. 

Thus,  if  F  and  F'  are  two  fixed 
points,  and  if  the  point  P  moves  about 
F  in  such  a  manner  that  the  sum  of  its 
distances  from  F  and  F'  is  always  the 
same,  the  point  P  will  describe  an  el- 
lipse, of  which  F  and  F'  are  the  foci. 
The  distance  of  the  point  P  from  either 
focus  is  called  the  radius  vector. 

(67.)  From  the  definition  of  an  ellipse,  the  curve  may  be  de- 
scribed mechanically.  Thus,  take  a  thread  longer  than  the 
distance  FF',  and  fasten  one  of  its  extremities  at  F,  the  other  at 
F'.  Then  let  a  pencil  be  made  to  glide  along  the  thread,  so  as 
to  keep  it  always  stretched  ;  the  curve  described  by  the  point 
of  the  pencil  will  be  an  ellipse. 

The  center  of  the  ellipse  is  the  middle  point  of  the  straight 
line  joining  the  foci. 

A  diameter  is  a  straight  line  drawn  through  the  center  and 
terminated  both  ways  by  the  curve. 

The  major  axis  is  the  diameter  which  passes  through  the 
foci.  The  minor  axis  is  the  diameter  which  is  perpendicular 
to  the  major  axis. 

The  parameter  of  the  major  axis  is  the  double  ordinate  which 
passes  through  one  of  the  foci. 

Proposition  I. — Theorem. 
(68.)   The  equation  of  the  ellipse,  referred  to  its  center  and 

axes,  is 

Ay  +  BV=A2B2, 

where  A  and  B  represent  the  semi-axes,  and  x  and  y  the  gen 

eral  co-ordinates  of  the  curve. 


On   the   Ej.lits 


57 


Let  Fand  F  be  the  foci,  and 
draw  the  rectangular  axes  CX, 
CY,  the  origin  C  being  placed 
at  the  middle  of  FF'.  Let  P 
be  any  point  of  the  curve,  and  A 
draw  PR  perpendicular  to  CX. 
Let  the  sum  of  the  distances  of 
the  point  P  from  the  foci  be 
represented  by  2A.  Denote  the  distance  CF  or  CF'  by  c ; 
FP  by  r,  and  F'P  by  r' ;  and  let  x  and  y  represent  the  co-or- 
dinates of  the  point  P. 

Then,  since  FF=PR2+RF, 


we  have  r2=yl+(x—  c)\ 

Also,  ,PF'2=PR2+RF'2; 

that  is,  r'*=y*  +  (x  +  cy. 

Adding  equations  (1)  and  (2),  we  obtain 

r*  +  r'*=2(y*+x*  +  c"-)  ; 

and,  subtracting  equation  (1)  from  (2),  we  obtain 

r'2— r3=4ca;, 

which  may  be  put  under  the  form 

(?-'  +  r)  (r'  —  r)  =  4cx. 

But  we  have,  from  the  definition  of  the  ellipse, 

r'  +  r=2A. 

Substituting  this  value  in  equation  (4),  we  obtain 

2c  x 

~  A  ' 

Combining  the  last  two  equations,  we  find 

ex 
r'=A+T 


(1) 
(2) 

(3) 
(4) 


and 


ex 

r=k — 7- 

A 


(5) 
(0) 


Squaring  these  values,  and  substituting  them  in  equation  (9) 
we  obtain 

A*+C^r=f+x-+c\ 


which  may  be  reduced  to 

Ay+(A2-cy=A5(A5-c'), 

which  is  the  equation  of  the  ellipse. 


(1) 


58 


Analytical    Geometry. 


This  equation  may,  however,  be  put  under  a  more  conven- 
ient form.     Represent  the  line  y 
BC  by  B.    In  the  two  right-an- 
gled triangles  BCF,  BCF',  CF 
is  equal  to  CF',  and  BC  is  com- 
mon to  both  triangles  ;  hence  •A- 
BF  is  equal  to  BF'.     But  BF 
+BF',  by  the  definition  of  the 
ellipse,  is  equal  to  2A ;  conse- 
quently BF  is  equal  to  A. 

Now  BC2=BF-FC2: 

that  is,  B2=A2-c2. 

Substituting  this  value  in  equation  (7),  we  obtain 

Ay+BV=A2B2, 
which  is  the  equation  required. 

(69.)  Scholium.  Transposing,  and  dividing  by  A2,  this  equa- 


(8) 


B2 


tion  reduces  to  ?/2=-t-j(A 

Cor.  1.  To  determine  where  the  curve  intersects  the  axis 
of  abscissas,  make  y—0  in  the  equation  of  the  ellipse,  and  we 
obtain 

z=±A=:CA  or  CA', 
which  shows  that  the  curve  cuts  the  axis  of  X  in  two  points, 
A  and  A',  at  the  same  distance  from  the  origin,  the  one  being 
to  the  right,  the  other  to  the  left ;  and,  since  2CA  or  AA'  is 
equal  to  2A,  it  follows  that  the  sum  of  the  two  lines,  drawn  from 
any  point  of  an  ellipse  to  the  foci,  is  equal  to  the  major  axis. 

Cor.  2.  If  we  make  x—0,  in  the  equation  of  the  ellipse,  we 
obtain 

y=±B=CBorCB', 

which  shows  tnat  the  curve  cuts  the  axis  of  Y  in  two  points, 
B  and  B',  at  the  same  distance  from  the  origin. 

Cor.  3.  When  B  is  made  equal  to  A,  the  equation  of  the 
ellipse  becomes 

i/2-f  .-c2=A2, 
which  is  the  equation  of  a  circle;  hence  the  ellipse  becomes  a 
circle  when  its  axes  are  made  equal  to  each  other. 

Cor.  4.  Since  BF  or  BF'  is  equal  to  A,  it  follows  that  the 


On    the   Ellipse.  69 

distance  from  either  focus  to  the  extremity  of  the  minor  axis,  is 
equal  to  half  the  major  axis. 

Cor.  5.  According  to  the  Scholium,  Art.  69, 

2/'=!kA2-*2). 
Suppose  x=c,  or  CF,  then 

But  by  Art.  68,  Equation  (8), 

A2-c2=B2. 

Hence  y*=—xB\ 

or  A:    B::    B:    y, 

and  2A  :  2B  : :  2B  :  2y. 

But  2y  represents  the  double  ordinate  drawn  through  the 
focus,  and  is  called  the  parameter,  Art.  67 ;  hence  the  parame 
ter  is  a  third  proportional  to  the  major  and  minor  axes. 

Cor.  6.  The  quantity  -r-»  or  the  distance  from  the  center  to 

A. 

either  focus,  divided  by  the  semi-major  axis,  is  called  the  eccen- 
tricity of  the  ellipse.  If  we  represent  the  eccentricity  by  e, 
then 

— =e,  or  c=Ae. 
A 

But  we  have  seen  that  c2=A2— B\ 

Hence  A2-B2=AV, 

B'     .      a 
AT1—" 

Making  this  substitution,  the  equation  of  the  ellipse  becomes 

V2=(l-e2)(A2-.<). 

Cor.  7.  Equations  (5)  and  (6)  of  the  preceding  Proposition 

are 

ex 

r=A+T 

*  cx 

r=A — r. 

A 

Substituting  e  for  -p  these  equations  became 


60 


Analytical  Geometry. 


r'=A+ex, 
r  =  A  —  ex, 
which  equations  represent  the  distance  of  any  point  of  the  el- 
lipse from  either  focus. 

Multiplying  these  values  together,  we  obtain 
rr'=A'  —  e2x-2, 
which  is  the  value  of  the  product  of  the  focal  distances. 


Proposition  II. — Theorem. 

(70.)   The  equation  of  the  ellipse,  when  the  origin  is  at  the 
vertex  of  the  major  axis,  is 

where  A  and  B  represent  the  semi-axes,  and  x  and  y  the  gen- 
eral co-ordinates  of  the  curve. 

The  equation  of  the  ellipse,  when       y 
the  origin  is  at  the  center,  is 

Ay+BV=A2B2.         (1) 

If  the  origin  is  placed  at  A',  the 
ordinates  will  have  the  same  value 
as  when  the  origin  was  at  the  center, 
but  the  abscissas  will  be  different. 

If  we  represent  the  abscissas  reckoned  from  A'  by  x',  then 
it  is  plain  that  we  shall  have 

CR=A'R-A'C, 
or  x=x'  —  A. 

Substituting  this  value  of  x  in  equation  (1),  we  have 
Ay+BV2-2ABV  =  0, 
which  may  be  put  under  the  form 

T>2 

f=j;(2Ax'-x")  ; 

or,  omitting  the  accents, 

R2 

y*=ji(i&x-x-), 

which  is  the  equation  of  the  ellipse  referred  to  the  vertex  oi 
the  major  axis  as  the  origin  of  co-ordinaies. 


On   the    Ellipse. 


G\ 


Proposition  III. — Theorkm. 

(71.)  The  square  of  any  ordinate  is  to  the  product  of  the 
parts  into  which  it  divides  the  major  axis,  as  the  square  of  the 
minor  axis  is  to  the  square  of  the  major  axis. 

The  equation  of  the  ellipse,  re- 
ferred to  the  vertex  A'  as  the  origin 
of  co-ordinates,  is,  Art.  70, 

Ba 

y^=ji(2A-x)x. 

This  equation  may  be  resolved 
into  the  proportion 

if  :  {2k-x)x  : :  B2  :  A2. 

Now  2A  represents  the  major  axis  AA',  and,  since  x  repre- 
sents A'R,  2 A— x  will  represent  AR  ;  therefore  (2 A— x)x  rep- 
resents the  product  of  the  parts  into  which  the  major  axis  is 
divided  by  the  ordinate  PR. 

Cor.  It  is  evident  that  the  squares  of  any  two  ordinates  are 
as  the  products  of  the  parts  into  which  they  divide  the  major 
axis. 

Scholium.  It  may  be  proved  in  a  similar  manner  that  the 
squares  of  ordinates  to  the  minor  axis  are  to  each  other  as  the 
products  of  the  parts  into  which  they  divide  the  minor  axis. 


Proposition  IV. — Theorem. 

(72.)  If  a  circle  be  described  on  the  major  axis  of  an  ellipse, 
then  any  ordinate  in  the  circle  is  to  the  corresponding  ordinate 
in  the  ellipse,  as  the  major  axis  is  to  the  minor  axis. 

If  we  represent  the  ordinate  PR 
in  the  ellipse  by  y',  and  the  ordinate 
P'R  in  the  circle  corresponding  to 
the  same  abscissa  A'R  by  Y',  the 
equation  of  the  ellipse  will  give  us,  a[ 
by  Art.  69, 

>/"=|(A2-*2), 

and  the  equation  of  the  circle  will 
give,  Art.  33, 

Y'2=(A3-.r2). 


62 


Analytical   Geometry. 


Combining  these  two  equations,  we  have 
?/'2=— -Y'2 

y    Aa*  » 


or 


y<: 


A*  ' 


whence  we  derive  the  proportion 

Y'  :y'  ::  A  :  B  : :  2A  :  2B. 

(73.)  Cor.  In  the  same  manner,  it  may  be  proved  that  if  a 
circle  be  described  on  the  minor  axis  of  an  ellipse,  any  ordinate 
drawn  to  the  minor  axis  is  to  the  corresponding  ordinate  in 
the  circle,  as  the  major  axis  is  to  the  minor  axis. 

If  we  represent  the  ordinate  PR  in 
the  ellipse  by  x',  and  the  correspond- 
ing ordinate  P'R  in  the  circle  by  X', 
we  shall  have,  Prop.  I, 

and  X'*=B2-y\ 

Combining  these  two  equations,  we  have 

A2 

X    —  jj-2A    , 


or 


B 


X' 


whence  we  derive  the  proportion 

x'  :  X'  : :  A  : :  B  : :  2A  :  2B. 


Proposition  V. — Theorem. 
f  (74.)  Every  diameter  of  an  ellipse  is  bisected  at  the  center. 

Let  PP'  be  any  diameter  of  an 
ellipse.  Let  x',  y'  be  the  co-ordi- 
nates of  the  point  P,  'and  x",  v" 
those  of  the  point  P'.  Then,  from 
the  equation  of  the  ellipse,  we  shall 
have.  Art.  69 

y'-|i(A9-a:rt), 
3r=4a(As~s"9); 


whence 


On   the   Ellipse. 

y"  _A'-x" 


63 


But  from  the  similarity  of  the  triangles  PCR,  P'CR'  we  have 


whence 


y'  _  x' 


x"2    A'-x"1' 
Clearing  of  fractions,  we  obtain 

„/2 ~//»  . 


whence,  also, 
Consequently, 
or 
that  is, 


x'  =  X' 

y'*=y"\ 


x>*+y"=x"2+y"\ 
CF=CP/a ; 
CP  =CF. 

Proposition  VI. — Theorem. 


(75.)  If  from  the  vertices  of  the  major  axis,  two  lines  be  drawn 
to  meet  on  the  curve,  the  product  of  the  tangents  of  the  angles 
which  they  form  with  it,  on  the  same  side,  will  be  negative,  and 
equal  to  the  square  of  the  ratio  of  the  semi-axes. 

The  equation  of  the  line  AP, 
passing  through  the  point  A, 
whose  co-ordinates  are  x'=A, 
y'=0,  Art.  18,  is 

y=a(x-A). 

The  equation  of  A'P,  passing 
through  the  point  A',  whose  co- 
ordinates are  x"  =  —  A,  y"=0,  Art.  18,  is 

y—a'{x+A). 
These  lines  must  pass  through  the  point  P  in  the  ellipse. 
Hence,  if  we  represent  the  co-ordinates  of  P  by  x"  and  y",  we 
have  the  three  equations 

y"  =a(x"-A) 
y"  =aV'+A)  . 

y'liJ^x-i-A*) 

Multiplying  (1)  and  (2)  together,  we  have 

y"2  =  aa'(x"2-A2). 
Hence,  comparing  with  (3),  we  see  that 


(1) 

(2) 

(3) 


G4 


Analytical   Geometky. 


(16.)  Scholium.  Two  lines  which  are  drawn  from  the  same 
point  of  a  curve  to  the  extremities  of  a  diameter,  are  called  sup- 
plementary  chords. 

Cor.  In  the  circle,  which  may  be  considered  an  ellipse 
whose  two  axes  are  equal  to  each  other,  we  have 

aa'—  —  1, 
which  shows  that  the  supplementary  chords  are  perpendicular 
to  each  other  (Art.  24). 

Proposition  VII. — Theorem. 

(77.)  The  equation  of  a  straight  line  which  touches  an  ellipse  is 
Ayij'+B\xx'=A*B\ 
where  x  and  y  are  the  general  co-ordinates  of  the  tangent 
line,  x'  and  y'  the  co-ordinates  of  the  point  of  contact. 

Draw  any  line,  P'P",  cutting 
the  ellipse  in  the  points  P',  P"  ;  if 
.his  line  be  moved  toward  P,  it 
will  approach  the  tangent,  and 
the  secant  will  become  a  tangent 
when  the  points  P',  P"  coincide. 

Let  x',  y'  be  the  co-ordinates 
of  the  point  P',  and  x",  y"  the  co-ordinates  of  the  point  P' 
The  equation  of  the  line  P'P",  passing  through  these  two  points 
Art.  20,  will  be 

y'-yr 


y-y'- 


7i(x-x')- 


(1) 


x'-x' 

Since  the  points  P',  P"  are  on  the  curve,  we  shall  have 
A'y"  +BV2  =A2B%  (2) 

and  Ay2+BV"=A!B2.  (3) 

Subtracting  equation  (3)  from  (2),  we  have 
A*(y'*-ij'/')+B\x'2-x,r-)  =  0, 
or  A'iy'-y")  {y'+y")  =  -B\x'-xn)  (x'+x"). 

tj'-y"  _     B2/x'+x"\ 
x'-x"~~A*\y'  +  tj")' 
Substituting  this  value  in  equation  (1),  the  equation  of  the 
secant  line  becomes 


Whence 


On   the   Ellitse. 
B'x'+x" 


65 


y-y'=- 


T&-X% 


(4) 


The  secant  P'P"  will  become  a  tangent  when  the  points  P'f 
P"  coincide,  in  which  case 

x'=x"  and  y'=y". 
Equation  (4),  in  this  case,  becomes 

B'x' 

y-y'=-jt^(x-x')> 

which  is  the  equation  of  a  tangent  to  the  ellipse  at  the  point  P. 
If  we  clear  this  equation  of  fractions,  we  have 

Aayy'-A*y»  =  -B\vx'+B'x,^2, 
or  A7yy'+B2xx'=Ay"+B\vr' *».  Jx^ 

hence  A'yij'+B'xx^A^B', 

which  is  the  most  simple  form  of  the  equation  of  a  tangent  line. 
(78.)   Cor.  1.  In  the  equation 


y-y 


Wx 

A*y 


0> 


B2  x' 


— -r-2—  represents  the  trigonometrical  tangent  of  the  angle 
A-y 

which  the  tangent  line  makes  with  the  major  axis. 

Cor.  2.    To   find   the   point   in 

which  the  tangent  intersects  the 

axis  of  abscissas,  make  y—0  in  the 

equation  of  the  tangent,  and  we 

have 

A2 
x—— • 
x' 

which  is  equal  to  CT. 

If  from  CT  we  subtract  CR  or  x',  we  shall  have  the  sub- 
tangent 


RT: 


A2-: 


x'  x' 

Cor.  3.  This  expression  for  the  subtangent  is  independent 
of  the  minor  axis  ;  the  subtangent  is,  therefore,  the  same  for  all 
ellipses  having  the  same  major  axis ;  it  consequently  belongs 
to  the  circle  described  upon  the  major  axis. 

Cor.  4.  Hence  we  are  enabled  to  draw  a  tangent  to  an  el° 
Hpse  through  a  given  point.     Let  P  be  the  given  point.     On 


66 


Analytical  Geometry. 


AA'  describe  a  circle,  and 
through  P  draw  the  ordinate 
PR,  and  produce  it  to  meet  the 
circumference  of  the  circle  in 
P'.  Through  P'  draw  the  tan- 
gent P'T,  and  from  T,  where  it 
meets  the  major  axis  produced, 
draw  PT  ;  it  will  be  a  tangent 
to  the  ellipse  at  P. 

Cor.  5.  Since  the  co-ordinates  of  the  point  P  are  equal  to 
those  of  the  point  P',  it  follows  from 
Cor.  1  that  the  tangents  at  the  ex- 
tremities of  a  diameter  make  equal 
angles  with  the  major  axis,  and  are 
therefore  parallel  with  each  other. 

Hence,    if    tangents     are     drawn 
through  the  vertices  of  any  two  di- 
ameters, they  will  form  a  parallelogram  circumscribing  the 
ellipse. 

Proposition  VIII. — Theorem. 
(79.)    The  equation  of  a  normal  line  to  the  ellipse  is 


y-y 


Ay 


where  x  and  y  are  the  general  co-ordinates  of  the  normal  line, 
x'  and  y'  the  co-ordinates  of  the  point  of  infovsection  with  the 
ellipse. 

The  equation  of  a  straight  line 
passing  through  the  point  whose 
co-ordinates  are  x',  y',  Art.  18,  is 
y-y'=a(x-x');  (l) 
and,  since  the  normal  line  is  per- 
pendicular to  the  tangent,  we  shall 
have,  Art.  23, 

1 


But  we  have  found  for  the  tangent  line,  Prop.  VII.,  Cor.  1, 

BV 

a'~  ~TT~> 

Ay 


Hence 


On    the   Ellipse. 

Ay 


67 


a— 


BV 


Substituting  this  value  in  equation  (1),  we  shall  have  for  the 
equation  of  the  normal  line 


(2) 


(80.)  Cor.  1.  To  find  the  point  in  which  the  normal  inter- 
sects the  axis  of  abscissas,  make  y=0  in  equation  (2),  and  we 
have,  after  reduction, 

A2-B2 

If  we  subtract  this  value  from  CR,  which  is  represented  by 
x',  we  shall  have  the  subnormal 

A2-B2       BV 

NR=z' jt-  z-^r- 

A2— B2 

Cor.  2.  If  we  put  e2  for  — -£-— ,  Art.  69,  Cor.  6,  we  shall  have 

CN=eV. 
If  to  this  we  add  F'C,  which  equals  c  or  Ac,  Prop.  I.,  Cor.  6, 
we  have 

F/N=Ae+eV=e(A+ca;')) 

which  is  the  distance  from  the  focus  to  the  foot  of  the  normal. 

Proposition  IX. — Theorem. 

(81.)   The  normal  at  any  point  of  the  ellipse  bisects  the  angle 
formed  by  lines  drawn  from  that  point  to  the  foci. 

Let  PT  be  a  tangent  line  to  an 
ellipse,  and  PF,  PF'  two  lines 
drawn  to  the  foci.  Draw  PN,  bi- 
secting the  angle  FPF'.  Then, 
by  Geometry,  Prop. XVII., B.  IV.,  ^~ 

FP:F'P::FN:F'N; 
or,  by  composition, 

FP+F'P  :  FF'  : :  F'P  :  FN.  (1) 

But  FP+F'P=2A. 

Also,  FF'  =  2c=2Ae,  Prop.  I.,  Cor.  6, 

and  F'P=A+er,  Prop.  I.,  Cor.  7. 

Making  these  substitutions  in  proportion  (1),  we  have 


68 


Analytical    Geometry 


2A  :  2Ae  : :  A+ex  :  F'N. 
Hence  F'N=e(A+e.r). 

But  by  Prop.  VIII.,  Cor.  2,  e(A+ex)  represents  the  distance 
from  the  focus  F'  to  the  foot  of  the  normal.  Hence  the  line 
PN,  which  bisects  the  angle  FPF',  is  the  normal. 

(82.)  Cor.  1.  Since  PN  is  perpendicular  to  TT',  and  the 
angle  FPN  is  equal  to  the  angle  F'PN,  therefore  the  angle 
FPT  is  equal  to  the  angle  F'PT' ;  that  is,  the  radii  vectores  are 
equally  inclined  to  the  tangent. 

Cor.  2.  This  proposition  affords  a  method  of  drawing  a  tan- 
gent line  to  an  ellipse  at  a  given  point  of  the  curve. 

Let  P  be  the  given  point;  draw 
the  radii  vectores  PF,  PF';  pro- 
duce PF'  to  G,  making  PG  equal 
to  PF,  and  draw  FG.  Draw  PT 
perpendicular  to  FG,  and  it  will 
be  the  tangent  required  ;  for  the 
angle  FPT  equals  the  angle  GPT, 
which  equals  the  vertical  angle  F'PT'. 

Proposition  X. — Theorem. 

(83.)  If,  through  one  extremity  of  the  major  axis,  a  chord  be 
drawn  parallel  to  a  tangent  line  to  the  curve,  the  supplementary 
chord  will  be  parallel  to  the  diameter  which  passes  through  the 
point  of  contact,  and  conversely. 

Let  DT  be  a  tangent  to  the 
ellipse,  and  let  the  chord  AP 
be  drawn  parallel  to  it ;  then 
will  A'P  be  parallel  to  the  W 
diameter  DD',  which  passes 
through  the  point  of  contact  D. 

Let  x',  y'  designate  the  co- 
ordinates of  D  ;  the  equation  of  the  line  CD  will  be,  Art.  15, 

y'=pa'x' ; 

V' 
whence  a'——. 

x' 

But,  by  Prop.  VII.,  Cor.  1,  the  tangent  of  the  angle  which 

the  tangent  line  makes  with  the  major  axis,  is 

BV 

a~ TT7- 

Ay 


On   the    Ellipse. 


G9 


Multiplying  together  the  values  of  a  and  a',  we  obtain 


aa  =  — 


5! 

A" 


which  represents  the  product  of  the  tangents  of  the  angles 
which  the  lines  CD  and  DT  make  with  CT. 

But  by  Prop.  VI.  the  product  of  the  tangents  of  the  angles 
Ba 


A2' 


TAT,  PA'A  is  equal  to  - 

Hence,  if  AP  is  parallel  to  DT,  A'P  will  be  parallel  to  CD, 
and  conversely. 

(84.)  Cor.  Let  DD'  be 
any  diameter  of  an  ellipse, 
and  DT  the  tangent  drawn 
through  its  vertex,  and  let 
the  chord  AP  be  drawn 
parallel  to  DT  ;  then,  by 
this  Proposition,  the  supple- 
mentary chord  A'P  is  parallel  to  DD'.  Let  another  tangent 
ET'  be  drawn  parallel  to  A'P,  it  will  also  be  parallel  to  DD'. 
Let  the  diameter  EE'  be  drawn  through  the  point  of  contact 
E;  then,  by  this  Proposition,  A'P  being  parallel  to  T'E,  AP  (and, 
of  course,  DT)  will  be  parallel  to  EE'.  Each  of  the  diameters 
DD',  EE'  is  therefore  parallel  to  a  tangent  drawn  through 
the  vertex  of  the  other,  and  they  are  said  to  be  conjugate  to 
one  another. 

Scholium.  Two  diameters  of  an  ellipse  are  said  to  be  con- 
jugate to  one  another,  when  each  is  parallel  to  a  tangent  line 
drawn  through  the  vertex  of  the  other. 

If  we  designate  by  a  and  a'  the  tangents  of  the  angles  which 
two  conjugate  diameters  make  with  the  major  axis,  then  we 
must  have 

B2 


Proposition  XI. — Theorem. 
(85.)   The  equation  of  the  ellipse,  referred  to  its  center  and 
conjugate  diameters,  is 

A'y+B'V=A'2B'2; 
vvhere  A'  and  B'  are  semi-conjugate  diameters. 


70 


Analytical   Geometry. 


The  equation  of  the  ellipse,  referred  to  its  center  and  axes, 
Art.  68,  is 

Ay+BV=A2B\ 
In  order  to  pass  from  rectangular  to  oblique  co-ordinates, 
the  origin  remaining  the  same,  we  must  substitute  for  x  and  y 
in  the  equation  of  the  curve,  Art.  30,  the  values 
x=x'  cos.  a+y'  cos.  a', 
y=x'  sin.  ct+y'  sin.  a'. 
Squaring  these  values  of  x  and  y,  and  substituting  in  the 
equation  of  the  ellipse,  we  have 


A2  sin.2  a' 
B2  cos.2  a 


y/2+2A2  sin.  a  sin.  a'lz'y'+A2  sin.2  a 

B2  cos.2  a 


.7j'2=A2B2;  (1) 


2B2  cos.  a  cos.  a' I 
which  is  the  equation  of  the  ellipse  when  the  oblique  co-ordi- 
nates make  any  angles  a,  a'  with  the  major  axis. 

But  since  the  new  axes  are  conjugate  diameters,  we  must 

have  (Art.  84) 

B2 

aa'= rr, 

A 

*  B2 

or  tang,  a  tang.  a'=  —  -^  ; 

whence  A2  tang,  t  tang.  a'+B2=0. 

Multiplying  by  cc.  a  cos.  a', 

remembering  that     cos.  a  tang.a=sin.  a, 
we  have        A2  sin.  a  sin.  a'+B2  cos.  a  cos.  a'=0. 
Hence  the  term  containing  x'y'  in  equation  (1)  disappears,  and 
we  have 

(A2sin.V+B2cos.V)3/'2  +  (A2sin.2  a+B2  cos.2«)x'2=A2B\  (2) 
which  is  the  equation  of  the  ellipse  referred  to  conjugate  diam- 
eters. 

If  in  this  equation  we  make  y'=0, 
we  shall  have 

A  sin.  a+B"  cos.  a 
•    If  we  make  x'  —  Q,  we  shall  have 
A2B2 


</'-- 


:CE2 


"A2  sin.2  a'+B2  cos.1 
If  we  represent  CD  by  A'  and  CE  by  B',  equation  (2)  re- 
duces to 


On   the   Ellipse. 


71 


y     x     , 

B'2    A'2 
hence  A'y9+B'Va=A'aB» ; 

or,  omitting  the  accents  from  x  and  y, 

A'y+B'V=A'2B'2, 
which  is  the  equation  of  the  ellipse  referred  to  its  center  and 
conjugate  diameters. 

Proposition  XII. — Theorem. 
(86.)    The  square  of  any  diameter  is  to  the  square  of  its  con- 
jugate, as  the  rectangle  of  the  parts  into  which  it  is  divided  by 
any  ordinate  is  to  the  square  of  that  ordinate. 
The  equation  of  the  ellipse,  referred  g 

to  conjugate  diameters,  is 

A'y+B'V=A'2B'2, 
which  may  be  put  under  the  form 
A'y=B'2(A'2-ar). 
This  equation  may  be  reduced  to  the 

proportion 

A'2  :  B'2  ::  A'2 -a;2  ■:  y\ 

or  (2A')2  :  (2B')2  : :  (A'+z)  (A'-x)  :  y\ 

Now  2A'  and  2B'  represent  the  conjugate  diameters  DD', 
EE';  and,  since  x  represents  CH,  A'+x  will  represent  D'H, 
and  A'-x  will  represent  DH  ;  also,  GH  represents  y ;  hence 
DD'2  :  EE'2  : :  DHxHD'  :  GH2. 
(87.)  Cor.  It  is  evident  that  the  squares  of  any  two  ordi- 
nates  to  the  same  diameter,  are  as  the  products  of  the  parts  into 
which  they  divide  that  diameter. 

Definition.  The  parameter  of  any  diameter  is  a  third  pro- 
portional to  that  diameter  and  its  conjugate. 

2B2 
The  parameter  of  the  major  axis  is  equal  to  — r-,  Art.  69,  Cor. 

2A2 
5,  and  that  of  the  minor  axis  to  --^-. 

Proposition  XIII. — Theorem. 
(88.)   The  sum  of  the  squares  of  any  two  conjugate  diameters 
is  equal  to  the  sum  of  the  squares  of  the  axes. 

Let  DD',  EE'  be  any  two  conjugate  diameters.     Designate 


72  Analytical   Geometry. 

the  co-ordinates  of  D  by  x',  y',  those  of  E  by  x",  y",  the  angle 

DCA  by  a,  and  the  angle  ECA  by  a'.  ^^__     _ 

Then, 

y' 

tang,  a  =-, 

y" 
tang-  a '--Tr 

v'v"  B2 

Therefore  tang,  axtang.  a  =^77>  which  equals  —ri,  be- 

cause  DD'  and  EE'  are  conjugate  diameters,  Prop.  X.,  Schol. 
Hence,  by  squaring  each  member  of  this  equation,  we  have 

Ayya=BW/B.  (1) 

But  because  the  points  D  and  E  are  on  the  curve,  we  have 

Ay2=A2B2-BV2, 

and  Ay/2=A2B2-BV2. 

Therefore,  by  multiplication, 

Ay Y /2= A4B4  -  A2B V2  -  A2B  V2 + B*x"x "2.    (2) 

Comparing  equation  (1)  with  equation  (2),  we  see  that 

A4B4-  A2BV2-  A2BV'2=0 ; 

or,  dividing  by  A2B4,  we  have 

A2-x"-x"'=0, 

or  A2=.r'2+.r"2.  (3) 

In  the  same  manner,  we  find  that 

B2=*/'2+y"2.  (4) 

Hence,  by  adding  equations  (3)  and  (4),  we  have 

A2+B2=:e,2  +  ?/'2+x-"2+?/"2=A'2+B'2. 

(89.)   Cor.  According  to  this  Proposition,  Equation  (3), 

x'^A'-x"*. 

Also,  from  the  equation  of  the  ellipse,  Art.  68, 

Ay2=B2(A2-a:"2). 

A2 
Hence  X'2=^V"\ 

or  s'  =  -gy". 

Tn  the  same  manner,  we  find 


B 

y'=jx". 


On   the   Ellipse. 


73 


Proposition  XIV. — Theorem. 

(90.)  If  from  the  vertex  of  any  diamster  straight  li?ies  are 
drawn  to  the  foci,  thei^  product  is  equal  ti  the  square  of  half  the 
conjugate  diameter. 

Represent  the  co-ordinates  of  the 
point  D,  referred  to  rectangular  axes, 
by  x',  y'. 

Then  the  square  of  the  distance  of 
D  from  the  center  of  the  ellipse  is 
An=xr-+ij'\ 

But  from  the  equation  of  the  curve,  Art.  68, 


A2 


Therefore,  by  substitution, 


A2 
=B2+eV2,  Art.  69. 
But,  by  Prop.  XIII.,  A'2+B'2=A2+B2. 


Therefore, 


B'2=A2-e2^ 


Also,  by  Prop.  I.,  Cor.  7, 


:A2- 


e'x' 


Hence 


'=B' 


Proposition  XV. — Theorem. 

(91.)  The  parallelogram,  formed  by  drawing  tangents  through 
the  vertices  of  two  conjugate  diameters,  is  equal  to  the  rectargle 
of  the  axes. 

Let  DED'E'  be  a  parallelo- 
gram, formed  by  drawing  tan- 
gents to  the  ellipse  through 
the  vertices  of  two  conjugate 
diameters  DD;,  EE';  its  area 
is  equal  to  AA'xBB'. 

Let  the  co-ordinates  of  D, 
referred  to  rectangular  axes, 
be  x',  y',  and  those  of  E  be  x",  y". 

The  triangle  CDE  is  equal  to  the  trapezoid  DEHG,  dimin- 
ished by  the  two  triangles  DCG,  EHC.     That  is, 


74  Analytical   Geometry. 

2.CDE  =  (x'+x")(y'+y")-x!y'-x"y", 
=x'y"Jrx"y', 

=*'x+2/'tp  hy  Pr°p- XIIL' Cor" 

BV2  +  Ay2  reducing  the  fractions  to  a  common 

A.B       '      denominator, 

A2B2 
=—-=-  =  A.B,  Art.  G8. 
A.B 

Therefore  the  parallelogram  CETD  is  equa  to  A.B  ;  and  the 

parallelogram  DED'E'  is  equal  to  4A.B  or  2Ax2B=AA'xBB'. 

Proposition  XVI. — Theorem. 
(92.)    The  polar  equation  of  the  ellipse,  when  the  pole  is  at  one 
of  the  foci,  is 

r= E , 

1+e  cos.  v 

where  p  is  half  the  parameter,  e  is  the  eccentricity,  and  v  is  the 

angle  which  the  radius  vector  makes  with  the  major  axis 

We  have  found  the  distance  of  any 

point  of  the  ellipse  from  the  focus, 

Prop.  I.,  Cor.  7,  to  be 

r=FP=A-ex, 

r'=F'P=A+ex, 

where    the    abscissa   x   is    reckoned 

from  the  center.     In  order  to  transfer 

the  origin  from  the  center  to  the  focus  F,  we  must  substitute 

for  x,  x'+c; 

or,  putting  Ae  for  c,  Art.  69,  we  have 

x=x'  +  Ae. 

If  we  represent  the  angle  PFA  by  v,  we  shall  have 

x'  =  r  cos.  v. 

Whence  x=r  cos.  v+Ae. 

Therefore  FP=r=A— er  cos.  v  —  Ae2. 

By  transposition,  r(l+e  cos.  v)  =  A— Ae2=A(l—  ea). 

•         2B*  /  A 

If  we  put  2p=  the  parameter  of  the  major  axis  =—  (Art. 

87),  Ave  shall  have 

p=A(l-e2),  Prop.  L,  Cor.  6. 

P 

Whence  r=— . 

1+e  cos.  v 


On    the   Ellipse. 


Proposition  XVII. — Theorem. 

(93.)  The  area  of  an  ellipse  is  a  mean  proportional  between 
the  two  circles  described  on  its  axes. 

Let  A  A'  be  the  major  axis  of  an 
ellipse  ABA'B'.  On  AA',  as  a  di- 
ameter, describe  a  circle  ;  inscribe 
in  the  circle  any  regular  polygon 
AM'MA',  and  from  the  vertices  M, 
M',  etc.,  of  the  polygon  draw  per-  -^j 
pendiculars  to  AA'.  Join  the  points 
B,  P,  etc.,  in  which  the  perpendic- 
ulars intersect  the  ellipse,  and  there 
will  be  inscribed  in  the  ellipse  a 
polygon  of  an  equal  number  of  sides. 

Let  Y,  Y'  be  the  ordinates  of  the  points  M,  M',  and  y,  y  the 
ordinates  of  the  points  P,  B,  corresponding  to  the  same  ab- 
scissas x,  x'. 

Y+Y' 
The  area  of  the  trapezoid  M'MRC= — (.-b— xO- 


The  area  of  the  trapezoid     BPRC= 
BPRC       y+y1 


-—  (x-x% 


Whence 
But,  by  Prop.  IV 

Whence 
consequently, 


M'MRC" 

B 
A 

y+y' 


Y+Y' 


2/=XY;  y'~ 


I- 


Y+Y' 
BPRC 


B 
A; 
B 
=  A' 


M'MRC 

In  the  same  manner  it  may  be  proved  that  each  of  ihe  trape- 
zoids composing  the  polygon  inscribed  in  the  ellipse,  is  to  the 
corresponding  trapezoid  of  the  polygon  inscribed  in  the  circle, 
in  the  ratio  of  B  to  A ;  hence  the  entire  polygon  inscribed  in 
the  ellipse,  is  to  the  polygon  inscribed  in  the  circle,  in  the  same 
ratio.     Hence,  if  we  represent  the  two  polygons  by  p  and  P, 

we  shall  have 

p_B 

P~A* 


76 


Analytical   Geometsy. 


Since  this  relation  is  true  whatever  be  the  number  of  sides 
of  the  polygons,  it  will  be  true  when  the  number  of  the  sides  is 
indefinitely  increased  ;  that  is,  it  is  true  for  the  ellipse  and  the 
circle,  which  are  the  limits  of  the  surfaces  of  the  polygons. 
Therefore,  if  we  represent  the  surfaces  of  the  ellipse  and  circle 
by  s  and  S,  we  shall  have 

•     B  B 

S  =  A'  01  5=SA- 
But  the  area  of  a  circle  whose  radius  is  A,  is  represented  b) 
ttA2  e  hence  the  surface  of  the  ellipse  is 

7rA2^  =  7rAB, 

A 

which  is  a  mean  proportional  between  the  two  circles  de- 
scribed on  the  axes.  For  the  area  of  the  circle  described  on 
the  major  axis  is  ttA2  ;  and  the  area  of  that  described  on  the 
minor  axis  is  7rB2 ;  and  7rAB  is  a  mean  proportional  between 
them. 

Proposition  XVIII. — Theorem. 

Any  chord  which  passes  through  the  focus  is  a  third  propor 
tional  to  the  major  axis  and  the  diameter  parallel  to  that  chord 

Let  PP'  be  a  chord  of  the  ellipse 
passing  through  the  focus  F,  and 
let  DD/  be  a  diameter  parallel  to 
PP'. 

By  Art.  92,  PF  =  r= — -£- . 

1+e  cos.  v 

H  we  substitute  for  v,  180°  -\-v,  we 
shall  have  the  value  of  P/F=r/= 


V 


Hence  we  have  PP 
By  Art.  85 
CD3= 


-.r+r'  = 


A3B2 


1  —  e  cos.  v 

2p 

1  — e3  cos.  2v 

A2B2 


A2  sin. 2v  +  B2  cos. 2v ~ A2  sin.  2v T(A2-AV)  cos.  V 

A2!^ A2(l-e3)   _  kp 

A2— A2e2  cos.  2v     1  —  e2  cos.2y~l— e3  cos.  V 
Hence  PP' :  CD3 :  :  2  :  A  ; 

n,  PP' :  2CD  :  :  CD  :  AC  ; 

and  AA' :  DD'  :  :  DD' :  PP'. 

This  property  includes  Cor.  5,  Prop.  I.,  page  59. 


SECTION   VII. 


ON   THE    HYPERBOLA. 

(94.)  An  hyperbola  is  a  plane  curve  in  which  the  difference 
of  the  distances  of  each  point  from  two  fixed  points  is  equal  to 
a  given  line.     The  two  fixed  points  are  called  the  foci. 

Thus,  if  F  and  F'  are  two  fixed 
points,  and  if  the  point  P  moves  about 
F  in  such  a  manner  that  the  difference 
of  its  distances  from  F  and  F'  is  al- 
ways the  same,  the  point  P  will  de- 
scribe an  hyperbola,  of  which  F  and 
F'  are  the  foci. 

If  the  point  P'  moves  about  F'  in 
such  a  manner  that  P'F— P'F'  is  always  equal  to  PF'  — PF,  the 
point  P'  will  describe  a  second  hyperbola  similar  to  the  first. 
The  two  curves  are  called  opposite  hyperbolas. 

(95.)  This  curve  may  be  described  by  continuous  motion  as 
follows : 

Let  F  and  F'  be  any  two  fixed 
points.  Take  a  ruler  longer  than 
the  distance  FF',  and  fasten  one  of 
its  extremities  at  the  point  F'. 
Take  a  thread  shorter  than  the 
ruler,  and  fasten  one  end  of  it  at  F, 
and  the  other  to  the  end  M  of  the 
ruler.  Then  move  the  ruler  MPF' 
about  the  point  F'.  while  the  thread  is  kept  constantly  stretch- 
ed by  a  pencil  pressed  against  the  ruler ;  the  curve  described 
by  the  point  of  the  pencil  will  be  a  portion  of  an  hyperbola. 
For,  in  every  position  of  the  ruler,  the  difference  of  the  lines 
PF,  PF'  will  be  the  same,  viz.,  the  difference  between  the 
length  of  the  ruler  and  the  length  of  the  string. 

If  the  ruler  be  turned,  and  move  on  the  other  side  of  the 
point  F,  the  other  part  of  the  same  hyperbola  may  be  described 


78 


Analytical   Geometry. 


Also,  if  one  end  of  the  ruler  be  fixed  in  F,  and  that  of  the 
thread  in  F',  the  opposite  hyperbola  may  be  described. 

(96.)  The  center  of  the  hyperbola  is  the  middle  point  of  the 
straight  line  joining  the  foci. 

A  diameter  is  a  straight  line  drawn  through  the  center,  and 
terminated  by  two  opposite  hyperbolas. 

The  transverse  axis  is  the  diameter  which,  when  produced, 
passes  through  the  foci. 

The  parameter  of  the  transverse  axis  is  the  double  ordinate 
which  passes  through  one  of  the  foci. 

Proposition  I. — Theorem. 

(97.)  The  equation  of  the  hyperbola,  referred  to  its  center 
and  axes,  is 

Ay-BV=-A2B2, 

where  A  and  B  represent  the  semi-axes,  and  x  and  y  are  the 
general  co-ordinates  of  the  curve. 

Let  F  and  F'  be  the  foci,  and 
draw  the  rectangular  axes  CX,  CY, 
the  origin  C  being  placed  at  the 
middle  of  FF'.  Let  P  be  any  point 
of  the  curve,  and  draw  PR  perpen- 
dicular to  CX.  Let  the  difference 
of  the  distances  of  the  point  P  from 
the  foci  be  represented  by  2A.  De- 
note the  distance  CF  or  CF  by  c,  FP  by  r,  F'P  by  r< ;  and  let 
x  and  y  represent  the  co-ordinates  of  the  point  P. 

Then,  since  FP2  =  PR2  +  RF, 

we  have  r=*/2  +  (r-c)\  (1) 

Also,  F'P2=PR2+RF'2; 

that  is,  r'2-?/2+(x+c)2.  (2) 

Adding  equations  (1)  and  (2),  we  obtain 

r°-  +  rr-= 2 (y+;r+c2)  ;  (3) 

and  subtracting  equation  (1)  from  (2),  we  obtain 

r'2— r^—^cx, 
which  may  be  written 

(r'  +  r)(r'-7-)  =  4cx.  (4) 

But,  from  the  definition  of  the  hyperbola,  we  have 
r'  —  r=2A. 


On   the   Hyperbola.  79 

Substituting  this  value  in  equation  (4),  we  obtain 

2cx 
r'  +  r=-r-. 
A 

Combining  the  last  two  equations,  we  find 

r'=     A+x,  (o) 

r=-A+f.  (G) 

Squaring  these  values  and  substituting  them  in  equation  (3), 

we  obtain 

c~x~ 

which  may  be  reduced  to 

Ay  +  (AJ-cs)f=A!(A!-c2)  (7) 

which  is  the  equation  of  the  hyperbola. 

If  we  put  B2  =  c2— A2,  the  equation  becomes 
Ay-BV=-A2B2, 
which  is  the  equation  required. 

(98.)  Scholium  1.  The  equation  of  the  hyperbola  differs  from 
that  of  the  ellipse  only  in  the  sign  of  B2,  which  is  positive  in 
the  ellipse,  and  negative  in  the  hyperbola. 

Transposing,  and  dividing  this  equation  by  A2,  it  may  be 
written 

y9=|kz9-Aa). 

Cor.  1.  To  determine  where  the  curve  intersects  the  axis 
of  abscissas,  make  y=0,  and  we  obtain 

a;=±A=CA  orCA', 
which  shows  that  the  curve  cuts  the  axis  of  X  in  two  points, 
A  and  A',  at  the  same  distance  from  the  origin,  the  one  being 
to  the  right,  and  the  other  to  the  left ;  and,  since  2CA  or  AA' 
is  equal  to  2A,  it  follows  that  the  difference  of  the  two  lines, 
drawn  from  any  point  of  an  hyperbola  to  the  foci,  is  equal  to  the 
transverse  axis. 

The  line  which  is  perpendicular  to  the  transverse  axis  at 
its  middle  point,  and  equal  to  2B,  is  called  the  conjugate  axis. 

Cor.  2.  When  B  is  made  equal  t<?   A.,  the  equation  of  the 

hyperbola  becomes 

y*-x*=-k\ 


80  Analytical   Geometry. 

In  this  case  the  hyperbola  is  said  to  be  equilateral 

Cor.  3.  Since  B2=c2  —  A2, 

and  A2+B2=c2  or  CF2, 

we  see  that  the  square  of  the  distance  from  the  center  to  either 
focus  is  equal  to  the  sum  of  the  squares  of  the  semi-axes. 

Cor.  4.  According  to  the  preceding  Scholium. 

In  order  to  determine  the  value  of  the  parameter  or  double 
ordinate  through  the  focus,  make  x— c  or  CF ;  then 

yB=£i(ca-<Aa). 

But  we  have  made        B2=c2— A2. 

"Da 

Hence  ?/2=-pXB2, 

or  A  :    B  : :    B  :    y, 

and  2A  :  2B  ::  2B  :  2y  ; 

that  is,  the  parameter  is  a  third  proportional  to  the  transverse 
and  conjugate  axes. 

c 
Cor.  5.  The  quantity  -r>  or  the  distance  from  the  center  to 

either  focus,  divided  by  the  semi-transverse  axis,  is  called  the 
eccentricity  of  the  hyperbola.  If  we  represent  the  eccentricity 
by  e,  then 

-r-=e,  or  c=Ae. 

But  we  have  seen  that 

B2=c2-A2. 

Hence  A2+B2=AV, 

B2      , 

Making  this  substitution,  the  equation  of  the  hyperbola  be 

comes  ?/2  =  (e2— 1)  (a;2  — A2). 

Cor.  6.   Equations  (5)  and  (6)  of  the  preceding  Proposition 

are 

ex 
>"=     A+j. 

cx 
r=-A+T 


On   the    Kvpfebola. 


81 


c 

Substituting  e  for  — ,  these  equations  become 

r'=ex  +  A, 

r  =  ex—A, 
which  equations  represent  the  distance  of  any  point  of  the 
hyperbola  from  either  focus. 

Multiplying  these  values  together,  we  obtain 

rr'=eV  — A2, 
which  is  the  value  of  the  product  of  the  focal  distances. 

Scholium  2.  If  on  BB',  as  a  trans- 
Verse  axis,  opposite  hyperbolas  are 
described  having  AA'  as  their  conju- 
gate axis,  these  hyperbolas  are  said  to 
be  conjugate  to  the  former. 

The  equation  of  the  conjugate  hyper- 
bolas may  be  found  from  the  equation 

Ay-BV=-A2Ba, 
by  changing  A  into  B  and  x  into  y.     It  then  becomes 

BV-Ay=-A2B2, 
which  is  the  equation  of  the  conjugate  hyperbolas. 


PaoposiTioN  II. — Theorem. 

(99.)   The  equation  of  the  hyperbola,  when  the  origin  is  a\ 
the  vertex  of  the  transverse  axis,  is 

tf=^(x*  +  2Ax), 

where  A  and  B  represent  the  semi-axes,  and  x  and  y  the  gen- 
eral co-ordinates  of  the  curve. 

The  equation  of  the  hyperbola,  when 

the  origin  is  at  the  center,  is,  Art.  97, 

Ay-BV=-A5B3.         (1) 

If  the  origin  is  placed  at  A,  the  or- 
dinates  will  have  the  same  value  as 
when  the  origin  was  at  the  center,  but 
the  abscissas  will  be  different. 

If  we  represent  the  abscissas  reckoned  from  A   by  x',  then 
it  is  plain  that  we  shall  have 


82  Analytical   Geometry 

CR=AR+AC, 

or  x=x'+A. 

Substituting  this  value  of  a;  in  equation  (1),  we  have 
Ay-BV8-2B2Az'=0, 
which  may  be  put  under  the  form 

y*=^(x»+2Ax') ; 

or,  omitting  the  accents, 

y*=-^(x>+2Az), 

which  is  the  equation  of  the  hyperbola  referred  to  the  vertex 
A  as  the  origin  of  co-ordinates. 

Proposition  III. — Theorem. 

(100.)  The  square  of  any  ordinate  is  to  the  product  of  its  dis- 
tances from  the  vertices  of  the  tranverse  axis,  as  the  square  of 
the  conjugate  axis  is  to  the  square  of  the  transverse  axis. 

The  equation  of  the  hyperbola,  re- 
ferred to  the  vertex  A  as  the  origin  of 
co-ordinates,  is,  Art.  99, 

This  equation  may  be  resolved  into 
the  proportion 

f  :  (.r  +  2A)x  :  :  B2  :  Aa. 

Now  2A  represents  the  transverse  axis  AA',  and,  since  x 
represents  A  R,  a:+2A  will  represent  A'R  ;  therefore,  (a;+2A)a; 
represents  the  product  of  the  distances  from  the  foot  of  the  or 
dinate  PR  to  the  vertices  of  the  transverse  axis. 

Cor.  It  is  evident  that  the  squares  of  any  two  ordinates  are 
as  the  products  of  the  parts  into  which  they  divide  the  trans- 
verse axis  produced. 

Proposition  IV. — Theorem. 

(101.)  Every  diameter  of  an  hyperbola,  is  bisected  at  the  center. 

Let  PP'  be  any  diameter  of  an  hyperbola.  Let  x',  y'  be  the 
co-ordinates  of  the  point  P,  and  x",  y '  those  of  the  point  P' 


On   the   Hyperbola.  8«J 

Then,  from  the  equation  of  the  curve,  we  shall  have,  Art.  9S 
Scholium  1, 

B2 

i 

and 


y"-=x-1(s»»--A'). 


y> 

Whence     — 


a;'2 -A3 


y-  V'2-A2' 

But  from  the  similarity  of  the  triangles  PCR,  P'CR',  we  have 
y'      x' 
y"     x  ' 

x"     a;'2 -A2 

Whence  — jjz=~ jjz — tz- 

x"     x    —A 

Clearing  of  fractions,  we  obtain 
x"=x"\ 
Whence,  also,  2/'2=*/"2- 

Consequently,  x»+y'*=x"*+y"\ 

or  CP2=CP'2 ; 

that  is,  CP  =CP'. 

Proposition  V. — Theorem. 
(102.)  If  from  the  vertices  of  the  transverse  axis,  two  lines  be 
drawn  to  meet  on  the  curve,  the  product  of  the  tangents  of  the  an- 
gles which  they  form  with  it,  on  the  same  side,  will  be  equal  to 
the  square  of  the  ratio  of  the  semi-axes. 

The  equation  of  the  line  AP  pass- 
ing through  the  point  A,  whose  co- 
ordinates are  x'=A,  y'=0,  Art.  18,  is 
y=a{x-A). 
The  equation  of  A'P  passing  through 
the  point  A',  whose  co-ordinates  are 
x'=-A,  y'  =  0,  Art.  18,  is 

y=a'(x+A). 
Tliese  lines  must  pass  through  the  point  P  in  the  hyperbola. 
Hence,  if  we  represent  the  co-ordinates  of  P  by  a?"  and  y"  we 
have  the  three  equations 

y"=:a(x"-A)  (1) 

y"  =  a'(x"  +  A)  (2) 


84 


Analytical   Geometry. 

Multiplying  (1)  and  (2)  together,  we  have 

Hence,  comparing  with  (3),  we  see  that 

B2 


(3) 


Cor.  In  the  equilateral  hyperbola  A=B,  and  we  have 
*     aa'=\; 
which  shows  that  the  angles  formed  by  the  supplementary 
chords,  with  the  transverse  axis  on  the  same  side,  are  together 
equal  to  a  right  angle,  Art.  24. 

Proposition  VI. — Theorem. 

(103.)   The  equation  of  a  straight  line  which  touches  an  hy 
perbola  is 

A\jij'-Wxx'=-A?W, 
where  x  and  y  are  the  general  co-ordinates  of  the  tangent  line, 
x'  and  y'  the  co-ordinates  of  the  point  of  contact. 

Draw  any  line  P'P"  cutting  the  hyper- 
bola in  the  points  P',  P";  if  this  line  be 
moved  toward  P  it  will  approach  the  tan- 
gent, and  the  secant  will  become  a  tan- 
gent when  the  points  P',  P"  coincide. 

Let  x',  y'  be  the  co-ordinates  of  the 
point  P',  and  x",  y"  the  co-ordinates  of 
the  point  P".  The  equation  of  the  line 
P'P",  passing  through  these  two  points, 
will  be,  Art.  20, 


V'-v" 
y-y'=  ,    y u(x-x'). 
y     y      x'—x"K  ' 


0) 


Since  the  points  P',  P"  are  on  the  curve,  we  shall  have, 
Art.  97, 

Ay2-BV2=-A2B2,  (2) 

Ay,2-BV2=-A2B2.  (3) 

Subtracting  equation  (3)  from  (2),  we  obtain 

A*(y'*-y"2)-B2(xl2-x"n-)  =  0, 
or  A*(y'+y")  {y> -y")-W(x'+x")  (x'-x")=0. 


Whence 


On   the   Hyperbola. 
y'-y"    W{x'+x) 


85 


x'-x"     A'iy'+y")' 
Substituting  this  value  in  equation  (1),  the  equation  of  the 
secant  line  becomes 

The  secant  P'P"  will  become  a  tangent  when  the  points  lv, 
P"  coincide,  in  which  case 

x'—x",  and  y'=y". 
Equation  (4)  in  this  case  becomes 

B*x' 

y-y  =j*-,(?-z'h 

which  is  the  equation  of  a  tangent  to  the  hyperbola  at  the  point 
P.     If  we  clear  this  equation  of  fractions,  we  obtain 

A'yi/  -  Ay2 = B'xx'  -  B  V2, 
or  Ay/'-B2;r:c'=-A2B2, 

which  is  the  most  simple  form  of  the  equation  of  a  tangent  line. 
(104.)   Cor.  1.  In  the  equation 

B2~' 


y-y 


A2*/ 


-,(z-x% 


B 


—  represents  the  trigonometrical  tangent  of  the  angle  which 
Ay 

Ihe  tangent  line  makes  with  the  transverse  axis. 

Cor.  2.  To  find  the  point  in  which  the 

tangent  intersects  the  axis  of  abscissas,  make 

y=0  in  the  equation  of  the  tangent  line,  and 

we  have 

A2 
x=— , 

x' 

which  is  equal  to  CT. 

If  from  CR  or  x'  we  subtract  CT,  we  shall 
have  the  subtangent 

x'         x' 

Proposition  VII. — Theorem. 
(105.)    The  equation  of  a  normal  line  to  the  hyperbola  is 

ay, 


y-!/'=-B^7(*-*')> 


86 


Analytical  Geometry. 


where  x  and  y  are  the  general  co-ordinates  of  the  normal  line 
and  x'  and  y'  the  co-ordinates  of  the  point  of  intersection  with 
the  curve. 

The  equation  of  a  straight  line  passing 
through  the  point,  whose  co-ordinates  are 
w\  y'f  Art.  18,  is 

y-y'=a(x-x') ;  (1) 

and,  since  the  normal  line  is  perpendicular    c    T 
to  the  tangent,  we  shall  have,  Art.  23, 
1 


a=- 


-a' 

But  we  have  found  for  the  tangent  line,  Prop.  VI.,  Cor.  1, 

BV 


a'= 


Hence 


a=  — 


AY 

A> 

BV 


Substituting  this  value  in  equation  (1),  we  shall  have  for  the 
equation  of  the  normal  line 


Ay 


(2) 


(106.)  Cor.  1.  To  find  the  point  in  which  the  normal  inter- 
sects the  axis  of  abscissas,  make  y=0  in  equation  (2),  and  we 
have,  after  reduction, 

l;N=z= — — — x'. 
A2 

If  we  subtract  CR,  which  is  represented  by  x',  we  shall  have 
he  subnormal 

PAT      A*  +  W    ,        ,      BV 
KiM  = — — x'  —  x'  =  —nr. 


Cor.  2.  If  we  put  e5 


A: 
A2+B: 


,  Art.  98,  Cor.  5,  we  shall  have 


A2 
CN=eV. 

If  to  this  we  add  F'C  (see  next  figure),  which  equals  c  or  Ae, 
Prop.  I.,  Cor.  5,  we  have 

F'N=Ae+eV=e(A+ca;')f 
which  is  the  distance  from  the  focus  to  the  foot  of  the  normal. 


On  the  Hyperbola. 


87 


Proposition  VIII.— Theorem. 
(107.)  A  tangent  to  the  hyperbola  bisects  the  angle  contained 
by  lines  drawn  from  the  point  of  contact  to  the  foci. 

Let  PT  be  a  tangent  line  to 
the  hyperbola,  and  PF,  PF'  two 
lines  drawn  to  the  foci.  Produce 
F'P  to  M,  and  draw  PN  bisect- 
ing the  exterior  angle  FPM. 

Then,  by  Geom.,  Prop.  XVII., 
Schol.,  B.  IV., 

F'P  :  FP  : :  F'N  :  FN  ; 
or,  by  division, 

F'P-FP  : :  F'F  :  :  F'P  :  FN.  (1) 

But  FP-FP-2A, 

F'F=2c  =  2Ae,  Prop.  I.,  Cor.  5, 
and  F'P=A  +ex,  Prop.  I.,  Cor.  6. 

Making  these  substitutions  in  proportion  (1),  we  have 
2A  :  2Ae  : :  A+ex  :  F'N. 
Hence  F'N=e(A+ez). 

But,  by  Prop.  VII.,  Cor.  2,  e(A+ex)  represents  the  distance 
trom  the  focus  F  to  the  foot  of  the  normal.  Hence  the  line 
PN  which  bisects  the  angle  FPM,  is  the  normal ;  that  is,  it  is 
perpendicular  to  the  tangent  TT'.  Now,  since  PN  is  perpen- 
dicular to  TT',  and  the  angle  FPN  is  equal  to  the  angle  MPN, 
therefore  the  angle  FPT  is  equal  to  MPT',  or  its  vertical  angle 
F'PT  ;  that  is,  the  tangent  PT  bisects  the  angle  FPF'. 

(108.)  Cor.  1.  The  normal  line  PN  bisects  the  exterior  an- 
gle FPM,  formed  by  two  lines  drawn  to  the  foci. 

Cor.  2.  This  Proposition  affords  a  method  of  drawing  a  tan- 
gent line  to  an  hyperbola  at  a  given 
point  of  the  curve.  4 

Let  P  be  the  given  point ;  draw 
the  radius  vectors  PF,  PF'.  On 
PF'  take  PG  equal  to  PF,  and  draw 
FG.  Draw  PT  perpendicular  to 
FG,  and  it  will  be  the  tangent  re- 
quired, for  it  bisects  the  angle 
FPF'. 


88 


Analytical   Geometry. 


Proposition  IX. — Theorem. 

(109.)  If  through  one  extremity  of  the  transverse  axis,  a  chord 
be  dr  awn  parallel  to  a  tangent  line  to  the  curve,  the  supplement- 
ary chord  will  be  parallel  to  the  diameter  which  passes  through 
the  point  of  contact,  and  conversely. 

Let  DT  be  a  tangent  to  the  hy- 
perbola, and  let  a  chord  AP  be 
drawn  parallel  to  it ;  then  will  A'P 
be  parallel  to  the  diameter  DD' 
which  passes  through  the  point  of 
contact  D. 

Let  x',  y'  designate  the  co-ordi- 
nates of  D;  the  equation  of  the  line 
CD  will  be,  Art.  15, 

y'=a'x' , 


whence 


a'  = 


V 


But,  by  Prop.  VI.,  Cor.  1,  the  tangent  of  the  angle  which  the 
tangent  line  makes  with  the  transverse  axis,  is 

BV 


a— 


Ay- 


Multiplying  together  the  values  of  a  and  a',  we  obtain 

B2 

which  represents  the  product  of  the  tangents  of  the  angles 
DCT  and  DTX.    But,  by  Prop.  V.,  the  product  of  the  tangents 

of  the  angles  PAX,  PA'A  is  also  equal  to  -t-j. 

A 

Hence,  if  AP  is  parallel  to  DT,  A'P  will  be  parallel  to  CD. 
and  conversely. 

(110.)  Let  DD'  be  any  diameter 
of  an  hyperbola,  and  DT  the^  tan- 
gent drawn  through  its  vertex,  and 
let  the  chord  AP  be  drawn  parallel 
to  DT  ;  then,  by  this  Proposition, 
the  supplementary  chord  A'P  is  par- 
allel to  DD'.  Let  another  tangent 
ET'  be  drawn  to  the  conjugate  hy- 
perbola parallel  to  A'P,  it  will  also 


On  the   Hyperbola.  8& 

be  parallel  to  DD'.  Let  the  diameter  EE'  be  drawn  through 
the  point  of  contact  E ;  then,  by  this  Proposition,  A'P  being 
parallel  to  T'E,  AP  (and,  of  course,  DT)  will  be  parallel  to 
EE'.  Each  of  the  diameters  DD',  EE'  is,  therefore,  parallel 
to  a  tangent  drawn  through  the  vertex  of  the  other,  and  they 
are  said  to  be  conjugate  to  one  another. 

Scholium.  Two  diameters  of  an  hyperbola  are  said  to  be 
conjugate  to  one  another,  when  each  is  parallel  to  a  tangent 
line  drawn  through  the  vertex  of  the  other. 

If  we  designate  by  a  and  a'  the  tangents  of  the  angles  which 
two  conjugate  diameters  make  with  the  transverse  axis,  then 

we  must  have 

B2 

aa'=-jTt. 

Proposition  X. — Theorem. 
(111.)   The  equation  of  the  hyperbola,  referred  to  its  center 
and  conjugate  diameters,  is 

A'y-B'V=-A'!B'2, 
where  A'  and  B'  are  semi-conjugate  diameters. 

The  equation  of  the  hyperbola,  referred  to  its  center  and 
nxes,  Art.  97,  is 

Ay-BV=-A2B2. 
In  order  to  pass  from  rectangular  to  oblique  co-ordinates, 
the  origin  remaining  the  same,  we  must  substitute  for  x  and  y, 
in  the  equation  of  the  curve,  Art.  30,  the  values 
x—x'  cos.  a,+y'  cos.  «,', 
y=x'  sin.  a+y'  sin.  a'. 
Squaring  these  values  of  x  and  y,  and  substituting  in  the 
equation  of  the  hyperbola,  we  have 

A2  sin.2  a'|?/'2+2A2  sin.  a  sin.  a'by  +  A2  sin.'  m/!=-  A2B2, 
--B2COS.2a'|      -2B2COS.  a  cos.  a'|         -B2  COS.2  a  (1) 

which  is  the  equation  of  the  hyperbola  when  the  oblique  co- 
ordinates make  any  angles  a,  a'  with  the  transverse  axis. 
But  since  the  new  axes  are  conjugate  'liameters,  we  must 

have,  Art.  110, 

B2 


90  Analytical    Geometry. 

B2 

or  tang,  a  tang.  a'=-ra- 

A. 

Whence  A2  tang,  a,  tang,  a'  —  B2=0. 

Multiplying  by  cos.  a  cos.  a.', 

remembering  that      cos.  a  tang.  a=sin.  a, 

we  have         A2  sin.  a  sin.  a'  — B2  cos.  a  cos.  a  .=0. 

Hence  the  term  containing  x'y'  in  equation  (1)  disappears,  and 

we  have 

(A2sin.2a'-B2cos.2a')y,2  +  (A2sin.2a-B2cos.2a)a;'2=-A2B2,  (2) 
which  is  the  equation  of  the  hyperbola  referred  to  conjugato 
diameters. 

If  in  this  equation  we  make  y'=0, 
we  shall  have 

A  sin.  a— 15-  cos.  a 
If  we  make  x'=0,  we  shall  have 

____A!B^__ 
y      A2sin.2a'-B2cos.2«' 

If  CD2  is  positive,  CE2  must  be  negative. 

For  when  CD2  is  positive,  the  numerator  of  the  above  ex- 
pression for  its  value  being  negative,  its  denominator  must  be 
negative ;  that  is, 

A2  sin.2  a<B2  cos.2  a, 

B2 
or  tang.'  a<—,  Trig.,  Art.  28. 

But  tang.2  a  tang.2  a'=T75  Art.  110. 

B2 

Hence  tang.2  a'>— , 

A 

or  A2  sin.2  a' >B2  cos.2  a'; 

that  is,  the  denominator  of  the  expression  for  CE2  is  positive; 

hence,  since  its  numerator  is  negative,  CE2  must  be  negative. 

If  we  represent  CD2  by  A'2,  and  CE2  by  -B2,  equation  (2) 

reduces  to 

V'2       x" 
— — A =4-1  • 

"D/2    '      A/2  '    *   » 

hence  A'y2-B'V2=-A'2B'2; 

or,  omitting  the  accents  from  x  and  y, 


On   the   Hyperbola.  91 

AY-B'V=-A"B", 
which  is  the  equation  of  the  hyperbola  referred  to  its  center 
and  conjugate  diameters. 

Proposition  XI. — Theorem. 

(112.)  The  square  of  any  diameter  is  to  the  square  of  its  con- 
jugate, as  the  rectangle  of  the  segments  from  the  vertices  of  the 
diameter  to  the  foot  of  any  ordinate,  is  to  the  square  of  that  or- 
dinate. 

The  equation  of  the  hyperbola,  re- 
ferred to  conjugate  diameters,  Art. 
Ill,  is 

A'y-B'V=-A'2B'2, 
which  may  be  put  under  the  form 
A'y=B'2(a;2-A'2). 

This  equation  may  be  reduced  to 
the  proportion 

A/2  :  B'2  : :  x2-A"  :  y\ 
or  (2A')2  :  (2B')2  : :  (ar+A')  (a;- A')  :  y\ 

Now  2A'  and  2B'  represent  the  conjugate  diameters  DD', 
EE'  ;  and,  since  x  represents  CH,  x-\-k!  will  represent  D'H, 
and  x— A'  will  represent  DH  ;  also,  GH  represents  y2 ;  hence 
DD'2  :  EE'2  : :  DHXHD'  :  GH2. 

(113.)  Cor.  It  is  evident  that  the  squares  of  any  two  ordinates 
to  the  same  diameter,  are  as  the  rectangles  of  the  correspond- 
ing segments  from  the  vertices  of  the  diameter  to  the  foot  of 
the  ordinates. 

Definition.  The  parameter  of  any  diameter  is  a  third  pro- 
portional to  the  diameter  and  its  conjugate.     The  parameter 

2B2 

of  the  transverse  axis  is  equal  to  — -r— ,  Art.  98,  Cor.  4  •  and  that 

2A2 
of  the  conjugate  axis  is  equal  to  —^-. 

Proposition  XII. — Theorem. 

(114.)    The  difference  of  the  squares  of  any  two  conjugate  di- 
ameters is  equal  to  the  difference  of  the  squares  of  the  axes. 
Let  DD',  EE'  be  any  two  conjugate  diameters.     Designate 


92 


Analytical   Geometry. 


the  co-ordinates  of  D  by  x',  y' ;  those  of  E  by  x",  y" ,•  the  an- 
gle  DCA  by  a,  and  the  angle  ECA  by 
*'.     Then 

y' 

tang,  a  =|j, 


tan 


I        j 


Therefore,  tang,  a  x  tang.  aJ~^-j—, 


x'x' 


B- 


which  equals  -r^,  because  DD'  and  EE'  are  conjugate  diame- 
ters, Prop.  IX.,  Schol. 

Hence,  by  squaring  each  member  of  this  equation,  we  have 

Ayy,2=BW2.  (1) 

But  because  the  point  D  is  on  the  curve,  we  have,  Art.  97, 
Ay2=-A2B2+BV2; 
and  because  E  is  on  the  curve  of  the  conjugate  hyperbola,  we 
have,  Art.  98,  Schol.  2, 

Ay/2=+A2B2+BV/2. 
Therefore,  by  multiplication, 

Ayy/2=  -  A4B4+A2BV2-  A2BV'2+BW'2.      (2) 
Comparing  equation  (1)  with  equation  (2),  we  see  that 
-A4B4+A2BV2-A2BV/2=0 ; 
or,  dividing  by  ~A2B4,  we  have 

A2-.-c/2+.r//2=0, 
or  A2=a;'2— x"\  (3) 

In  the  same  manner,  we  find  that 

B>=y>'*-y».  (4) 

Hence,  by  subtraction, 

A2-B2=a;'2+z/'2-;c"2-?/"2=A/2-B'2. 
(115.)   Cor.  According  to  this  Proposition,  Equation  (3), 

x'2=Ai+x"\ 
Also,  from  the  equation  of  the  hyperbola,  Art.  98,  Schol.  2f 
Ay2=B2(A2+z"2).     . 


Therefore 


A2 


On   the    Hyperbola. 


93 


or 


A 


In  the  same  manner  we  find 

B 

A' 


Proposition  XIII. — Theorem. 

(1 1 6.)  The  parallelogram  formed  by  drawing  tangents  througn 
the  vertices  of  two  conjugate  diameters,  is  equal  to  the  rectangh 
of  the  axes. 

Let  DED'E'  be  a  parallelogram 
formed  by  drawing  tangents  to  the 
hyperbola  through  the  vertices  of  two 
conjugate  diameters  DD',  EE' ;  its 
area  is  equal  to  AA'xBB'. 

Let  the  co-ordinates  of  D,  referred 
to   rectangular  axes,  be  x',  y' ;   and   /' 
those  of  E  be  x",  y". 

The  triangle  CDE  is  equal  to  the  trapezoid  DEHG,  plus  the 
triangle  ECH,  minus  the  triangle  CDG  ;  that  is, 
2.CDE=(x'-x")(tj'+y")+x"y"-x>y', 
~x'y"—x"y', 

=a.&L„^!f  by  Prop.  XIL,  Cor., 
A  U 

BV2  —  Ay2  reducing  the  fractions  to  a  common 

—       A.B        '      denominator, 

A2R2 
=x]r  A.B,  Art.  97. 

Therefore  the  parallelogram  CETD  is  equal  to  A.B ;  and  the 
parallelogram  DED'E'  is  equal  to  4  A.B  or  2  A  X  2B  =  A  A'  X  BB' 

Proposition  XIV. — Theorem. 
(117.)    The  polar  equation  of  the  hyperbola,  when  the  pole  is 
at  one  of  the  foci,  is 


1+e  cos.  vy 
where  p  is  half  the  parameter,  e  is  the  eccentricity,  and  v  is  the 
angle  which  the  radius  vector  makes  with  the  transverse  axis 


94 


Analytical   Geometry. 


We  have  found  the  distance  of 
any  point  of  the  hyperbola  from  the 
focus,  Prop.  L,  Cor.  6,  to  be 
r=FP=-A+e^, 
r'=F'P=     A+ex, 
where  the  abscissa  x  is  reckoned 
from  the  center.     In  order  to  trans- 
fer the  origin  from  the  center  to  the  focus  F,  we  must  substi 
tute  for  x,  x'+c;  or,  putting  Ae  for  c,  Art.  98,  Cor.  5,  we  have 

x=x'  +  Ae. 
If  we  represent  the  angle  PFC  by  v,  we  shall  have,  Art.  64, 
x'=—r  cos.  v. 
Whence  x=  —  r  cos.  v+Ae. 

Therefore,  FP=r=— A— er  cos.  v+Ae\ 

By  transposition, 

r(l+e  cos.  v)  =  -A+Ae*=A(e*-l). 

2B3 

If  we  put  2p=  the  parameter  of  the  transverse  axis  =-r=- 

A 

(Art.  113),  we  shall  have 

p=A(e*-l),  Prop.  I.,  Cor.  5. 

Whence  r= , 

1+e  cos.  v 

where  the  angle  v  is  estimated  from  the  vertex. 

ON  THE  ASYMPTOTES  OF  THE  HYPERBOLA. 

(118.)  If  tangents  to  four  conjugate  hyperbolas  be  drawn 
through  the  vertices  of  the  axes,  the  diagonals  of  the  rectangle 
so  formed,  supposed  to  be  indefinitely  produced,  are  called 
asymptotes  of  the  hyperbola. 

Let  AA',  BB'  be  the  axes  of 
four  conjugate  hyperbolas,  and 
-hrough  the  vertices  A,  A',  B, 
B',  let  tangents  to  the  curve  be 
drawn,  and  let  DD',  EE'  be  the 
diagonals  of  the  rectangle  thus 
formed ;  DD',  EE'  are  called 
asymptotes  to  the  curve. 

If  we  represent  the  angle 
DCX  by  a,  and  the  angle  E'CX 
by  a',  then  we  shall  have 


On   the   Asymptotes   of   the  Hyperbola. 

B 


95 


tang,  a  = 
taner.  a'  =  — 


A 
B 
A* 


But,  since  tang,  a- 


sin.  a 


,  Trig.,  Art.  28,  we  have 


or 


COS.  a 

sin.2  «_B2 

COS.2  a_  A2' 

sin.2  a    _B2 
1— sin.2  a- A2' 


Whence  sin.2  a= 

In  the  same  manner,  we  find 


B2 


A2+B2' 

A2 


COS.    "  =  J^tf> 

which  equations  furnish  the  value  of  the  angle  which  the  as 
ymptotes  form  with  the  transverse  axis. 

Proposition  XV. — Theorem. 
(119.)   The  equation  of  the  hyperbola,  referred  to  its  center 

and  asymptotes,  is 

A2+B2 

*y=— 4— ' 

where  A  and  B  are  the  semi-axes,  and  x  and  y  the  co  ordi- 
nates  of  any  point  of  the  curve. 

The  equation  of  the  hyperbola, 
referred  to  its  center  and  axes, 
Art.  97,  is 

Ay-BV=-AsBa.      (1) 
The  formulas  for  passing  from 
rectangular    to   oblique    co-ordi- 
nates, the  origin  remaining  the 
same,  Art.  30,  are 

x=x'  cos.  a+y'  cos.  a', 

y=x'  sin.  <*+?/'  sin.  a'. 

But,  since  a=— a',  these  equations  become 

z=(x'+y')  cos.  a, 

y=(x  —y')  sin  a 


OQ  Analytical   Geometry. 

Substituting  these  values  in  equation  (1),  we  have 
A:{x'-y'Y  sin.2  a-B'(x'+y'Y  cos.2  ct=--A2B2. 

R2 

But  sin.2  a=  ,  Art.  118, 

A  +r> 

and  cos.  a 


A2+B2' 
hence         j^(z'-yT-j^{x'+y>y=-tfW  5 

4A2B2 
that  is,  ABaa:y =A9B8, 

A2+B2 


'/,,/. 


or  arv — 

?  4 

which  is  the  equation  of  the  hyperbola  referred  to  its  center 
and  asymptotes. 

(120.)  Cor.  The  curve  of  the  hyperbola  approaches  nearer  the 
asymptote  the  further  it  is  produced,  but,  being  extended  ever  so 
fa?',  can  never  meet  it. 

The  equation  of  the  hyperbola,  referred  to  its  asymptotes, 
Art.  119,  is 

A2  +  B2 

A2+B2 
Put  Ma  for  — - — ,  and  we  have 

xy=W, 

Ma 
or  y= — ; 

J      x 

and,  since  M2  is  a  constant  quantity,  y  will  vary  inversely  as  x. 
Therefore  y  can  not  become  zero  until  x  becomes  infinite  ;  that 
is,  the  curve  can  not  meet  its  asymptote  except  at  an  infinite 
distance  from  the  center.  The  asymptotes  are,  therefore,  con- 
sidered as  tangent  to  the  curve  at  an  infinite  distance  from  the 
center. 

Proposition  XVI. — Theorem. 

(121.)  If  from  any  point  of  the  hyperbola  lines  be  drawn  par- 
allel to  and  terminating  in  the  asymptotes,  the  parallelogram  so 
formed  will  be  equal  to  one  eighth  the  rectangle  described  on  the 
axes. 

Designate  the  co-ordinates  of  the  point  P  referred  to  the- 


On   the   Asymptotes   of   the   Hyperbola.       97 

asymptotes  by  x',  y',  and  the  an- 
gle DCE',  included  between  the 
.asymptotes,  by  fi,  we  shall  have, 
from  the  equation  of  the  curve, 
Art.  119, 

A2+B2 

x'y'  sin.  (3= — - —  sin.  |3. 

The  first  member  of  this  equa- 
tion represents  the  parallelogram 
PC  contained  by  the  co-ordinates  of  the  point  P  of  the  curve. 

But  since  this  equation  is  true  for  every  point  of  the  curve, 
it  must  be  true  when  the  point  is  taken  at  the  vertex  A,  in 
which  case  a/  represents  CK,  and  y'  represents  AK,  and  we 
shall  have 


the  parallelogram 


A24-B2 
CHAK=      \      sin./3. 


Whence  the  preceding  equation  becomes 

x'y'  sin.  /3  =  the  parallelogram  CHAK. 

Therefore  the  parallelogram  PC,  formed  by  the  co-ordinates 
of  any  point  of  the  curve,  is  equal  to  the  parallelogram  HK, 
which  is  one  fourth  of  the  parallelogram  ABA'B',  or  one  eighth 
of  the  rectangle  described  on  the  axes. 


Proposition  XVII. — Theorem. 

(122.)   The  equation  of  a  tangent  line  to  an  hyperbola,  re 
ferred  to  its  center  and  asymptotes,  is 

V1 

y-y'=--,{x-x'), 

where  x',  y'  are  the  co-ordinates  of  the  point  of  contact. 

The  equation  of  a  secant  line  passing  through  the  points  x\ 
y\  x",  y",  Art.  20,  is 


y-y 


,   y'-y",      ,* 

'=— -{x— x'). 


(1) 


Since  the  two  given  points  are  on  the  curve,  we  must  have* 
Art.  120,  , 

x'y'=W, 
x"y"=W. 


Whence  x'y'=xry". 

G 


98 


Analytical  Geometry. 


Subtracting  x'y"  from  each  member,  we  have 
x'y,—x'y"=x"y"—x'y". 
Whence  x'{y'-y")=-y"(x'-x")% 

y'-y"  _   y"_ 

or  x'—x"         x' ' 

Hence,  by  substitution,  equation  (1)  becomes 

If  we  suppose  x'=x",  and  y'=y",  the  secant  will  become  a 
tangent,  and  equation  (2)  will  be 

y-y'=-v-te-x'h 

which  is  the  equation  of  the  tangent  line. 

(123.)  Cor.  To  find  the  point  in  which 
the  tangent  meets  the  axis  of  abscissas, 
make  y=0  in  the  equation  of  the  tangent 
line,  and  we  have 

x=2x' ; 
that  is,  the  abscissa  CT'  of  the  point,  where 
the  tangent  meets  the  asymptote  CE,  is 
double  the  abscissa  CM  of  the  point  of  tan- 
gency.  Therefore  CM=MT';  and,  since 
the  triangles  TCT',  PMT'  are  similar,  the 
tangent  TT'  is  bisected  in  P,  the  point  of  contact;  that  is,  if  a 
tangent  line  he  drawn  at  any  point  of  an  hyperbola,  the  part  in- 
cluded between  the  asymptotes  is  bisected  at  the  point  oftangency. 

Proposition  XVIII. — Theorem. 

(124.)  If  a  tangent  line  be  drawn  at  any  point  of  an  hyper- 
bola, the  part  included  between  the  asymptotes  is  equal  to  the  di- 
ameter which  is  conjugate  to  that  which  passes  through  the  point 
of  contact. 

Let  TT'  be  a  line  touching  the 
hyperbola  at  P.  Through  P  draw 
the  diameter  PP',  and  designate 
the  angle  contained  by  the  asymp- 
totes by  /3. 

Then,  by  Trigonometry,  Art. 
78,  in  the  triangle  CPM,  we  have 


On   the   Asymptotes   of   the   Hyperbola.       99 
CM2+MF-CP 


or 


cos.  CMP= 


—  cos.  ]3— 


2CMXMF 

x2+y2-CY2 


2xy 


Also,  in  the  triangle  PMT', 


cos.  PMT' 


or 


cos.  (3— 


PM2+MT'2-PT/!I 

2PM  XMT' 
x2+y2-VT'2 


2xy 


Whence  we  have 

C?2=x2+y2+2xij  cos.  (3, 
TT'2=x2+y2-2xy  cos.  j3. 
Whence  CP2-~PT'2=4xy  cos.  (3. 

But,  since  (3=2a,      cos.  (3=cos.2  a— sin.2  a,  Trig.,  Art.  74. 

A2-B2 

Hence,  from  Art.  118,  cos.  (3=  .  2     p2. 

Also,  from  the  equation  of  the  hyperbola,  Art.  119, 

A2+B2 

oi  4^=A2+B2. 

Therefore  4xy  cos.  |3=A2-B2, 

and  CP2-PT'2=A2-B2=A'2-B'2  (Prop.  XII.). 

But  CP  is  equal  to  A' ;  therefore  PT'=B' ;  that  is,  the  tan- 
gent TT'  is  equal  to  the  diameter  which  is  conjugate  to  PP'. 

(125.)  Cor..  The  same  is  true  of 
a  tangent  W  drawn  through  the 
point  P'  of  the  opposite  hyperbola. 
Therefore,  if  we  join  the  points  Tt, 
T't',  the  figure  Ttt'T'  will  be  a  par- 
allelogram whose  sides  are  equal 
and  parallel  to  2A',  2B' ;  that  is, 
PP',  EE'.  Hence  the  asymptotes 
are  the  diagonals  of  all  the  paral- 
lelograms which  can  be  formed  by  drawing  tangent  lines  through 
the  vertices  of  conjugate  diameters. 


SECTION   VIII. 


CLASSIFICATION  OF  ALGEBRAIC  CURVES. 

(126.)  We  have  seen  that  the  equations  of  the  circle,  the 
ellipse,  parabola,  and  hyperbola  are  all  of  the  second  degree ; 
we  will  now  show  that  every  equation  of  the  second  degree  is 
geometrically  represented  by  one  or  other  of  these  curves. 

The  general  equation  of  the  second  degree,  between  two 

variables,  is 

Ay*+Bxy+Cx*+T)y+Ex+F=0,        (1) 

which  contains  the  first  and  second  powers  of  each  variable, 
their  product,  and  an  absolute  term. 

Proposition  I. — Theorem. 

(127.)  The  term  containing  the  product  of  the  variables  in  tne 
general  equation  of  the  second  degree,  can  always  be  made  to  dis- 
appear, by  changing  the  directions  of  the  rectangular  axes. 

In  order  to  effect  this  transformation,  substitute  for  x  and  y, 
in  equation  (1),  the  values 

x—x'  cos.  a— y'  sin.  a,  )  ^) 

y—x''  sin.  a+y'  cos.  a,  ) 
by  which  we  pass  from  tne  system  of  rectangular  co-ordinates, 
to  another  having  the  same  origin,  Art.  29.     The  result  of  this 
substitution  is 

'+2Asin.acos.a' 
+Bcos.2a 
—  Bsin.2a 
—2C  sin.  a  cos.  a 
+D  cos.  a)  ,  (  +D  sin.  «. 
—  E  sin.  a  )  (  +E  cos.  a 

Since  the  value  of  a  is  arbitrary,  we  may  assume  it  of  such 
value  that  the  second  term  of  the  transformed  equation  may 
vanish.     We  shall  therefore  have 

2A  sin.  a  cos.  a+B  cos.2  a— B  sin.2  a— 2C  sin.  a  cos.  a=0. 
or  (A-C)2  sin.  a  cos.  a+B(cos.2  a-sin.2  a)=0. 


A  cos.2  on 

—  Bsin.acos.aiy/2+ 

C  sin.2aj 


!+Asin.2ai 
+Bsin.a  cos.  a>:c'* 
+Ccos.2aJ 


'+F=0. 


Classification   of  Algebraic  Curves.       103 

Bat  2  sin  a  cos.  a=sin.  2a, 

and  cos."  a— sin.2  a=cos.  2a,  Trig.,  Art.  74. 

Hence  (A  — C)  sin.  2a+B  cos.  2a=0  ; 

or,  dividing  by  cos.  2a, 

B 
tang.  2a=-JZIc' 

If,  therefore,  in  equations  (2),  we  give  to  the  angle  a  such  a 
value  that  the  tangent  of  double  that  angle  may  be  equal  to 

"D 

—  — — p,  the  term  containing  xy  will  disappear  from  the  trans- 

formed  equation.     The  new  equation,  therefore,  becomes  of 
the  form 

Mt/+Nx2+Ry+Sx+F=0.  (3) 

Proposition  II. — Theorem. 

(128.)  The  terms  containing  the  first  power  of  the  variables 
in  the  general  equation  of  the  second  degree,  can  be  made  to  dis- 
appear by  changing  the  origin  of  the  co-ordinates. 

In  order  to  effect  this  transformation,  substitute  for  x  and  y, 
in  equation  (3),  the  values 

x=a+x', 
y=b+y', 
by  which  we  pass  from  one  system  of  axes  to  another  system 
parallel  to  the  first,  Art.  28. 

The  result  of  this  substitution  is 
My,2+'Nxr-+2Mb  )  y'+2Na  )  :c'+M&2+Na2+R&+Sa+F=0. 
R    I  S     I 

In  order  that  the  terms  containing  x'  and  y'  may  disappear, 
ive  must  have 

T> 

2M6+R=0,  or  b=-^t 

S 
and  2Na+S=0,  or  a=  —  -^, 

where  a  and  b  are  the  co-ordinates  of  the  new  origin. 

If  we  employ  these  values  of  a  and  b,  and  substitute  P  foi 
-Mb2— Na2— Rb— Sa  —  F,  equation  (3)  reduces  to 

My*+Nx*=F, 
an  equation  from  which  the  terms  containing  the  first  power 
of  the  variables  have  been  removed.  . 


102  Analytical   Geometry. 

(129.)  If  one  of  the  terms  containing  x*  or  y2  was  wanting 

from  equation  (3),  this  last  result  would  be  somewhat  modified. 

If,  for  example,  N=0,  the  value  of  a,  given  above,  would  re- 

S 
duce  to  — ,  or  infinity.     We  can,  however,  in  this  case  cause 

the  term  which  is  independent  of  the  variables  to  disappear. 
For  this  purpose  we  must  put 

Mb2+Rb+Sa+F=0 

Mi2+R&+F 
which  gives  a— ~ . 

R 

With  this  value  of  a,  and  the  value  of  b=—  —^rr,  equation  (3) 

reduces  to  the  form 

Mj/2  +  Sx=0; 

S 
or,  putting  Q  for  -^,         2/*=Qz. 

Hence  every  equation  of  the  second  degree  between  two  varia- 
bles may  be  reduced  to  one  of  the  forms, 

My2  +  Nx2=P,  (4) 

or  y2=Qz.  (5) 

(130.)  Equation  (4)  characterizes  a  circle,  an  ellipse,  or  ar 
hyperbola. 

First.  Suppose  M,  N,  and  P  are  positive. 

P  P 

Put  A2=^,  and  B2=M. 

By  substituting  these  values  in  equation  (4),  we  obtain 
Vy2    Px2     _ 

— - — I =P 

B2  ^  A2         ' 

or  Ay+BV=A9B9, 

which  is  the  equation  of  an  ellipse,  Art.  68. 
If  M=N,  this  equation  characterizes  a  circle. 
Secondly.  If  N  and  P  are  both  negative,  or  the  equation  is  oi 
the  form 

Mya-Ns9=-P, 
P  P 

put  A2=N'  and  B  =M' 

and  we  obtain,  by  substitution, 


Classification   of   Algebraic  Curves.       103 

iy_iv 

Ba      A2  ~       ' 
01  Ay-BV=-A2B2, 

which  is  the  equation  of  an  hyperbola,  Art.  97. 

Thirdly.  If  N  alone  is  negative,  or  the  equation  is  of  the  form 
My2-Nx2=~P, 
we  shall  obtain,  by  substitution,  as  before, 

Ay-BV=A2B2, 
which  characterizes  the  conjugate  hyperbola,  Art.  98,  Schol.  2. 
(131.)  Equation  (5)  characterizes  a  parabola,  since,  by  put- 
ting Q=2p,  it  becomes 

if=2px,  Art.  50. 
Hence  the  only  curves  whose  equations  are  of  the  second  de- 
gree, are  the  circle,  parabola,  ellipse,  and  hyperbola. 

(132.)  When  the  origin  of  co-ordinates  is  placed  at  the  ver- 
tex of  the  major  axis,  the  equation  of  the  ellipse  is,  Art.  70, 

T>2 

tf  =  j;(2Ax-X>). 

The  equation  of  the  parabola  for  a  similar  position  of  the 
origin  is,  Art.  50, 

if=2px; 

and  the  equation  of  the  hyperbola  is,  Art.  99, 

T>2 

rf=-^(2Ax+x-). 

The  equation  of  the  circle  is 

y'=2llx-x\ 
These  equations  may  all  be  reduced  to  the  form 
y*=mx-\-nx*. 

In  the  ellipse,  m=-r-,  and  w= — -r. 

A  A 

In  the  parabola,         m=2p,  and  n=0. 

In  the  hyperbola,       m=—r-,  and  n—-r^. 
'  l  A  A2 

In  each  case  m  represents  the  parameter  of  the  curve,  and 
n  the  square  of  the  ratio  of  the  semi-axes.  In  the  ellipse,  n  is 
negative  ;  in  the  hyperbola  it  is  positive  ;  and  in  the  parabola  it 
is  zero. 


104         Analytical  Geometry. 

(133.)  Lines  are  divided  into  different  orders,  according  to 
the  degree  of  their  equations. 

A  line  of  the  first  order  has  its  equation  of  the  form 
Ay+Bx+C=0; 
this  class  consists  of  the  straight  line  only. 

Lines  of  the  second  order  have  their  equations  of  the  form 

Ay2 + Bxy + Cx* + By + Ex + F = 0. 
This  order  comprehends  four  species,  viz.,  the  circle,  ellipse, 
parabola,  and  hyperbola. 

(134.)  Lines  of  the  third  order  have  their  equations  of  the 
form 
Ay" + By'x  -f  Cyxa + LV + Eyn- + Fyx + Gx* +Hy +Ka;+L=0. 
Newton  has  shown  that  all  lines  of  the  third  order  are  com 
Drehended  under  some  one  of  these  four  equations, 
(1.)  xy'+Ey  =  Ax*+Bxi+Cx+V, 
(2.)  xy  =  Ax' + Bx* + Cx + D, 

(3.)  t/2=A:c3  +  B.r2  +  Cz+D, 

(4.)  y  ^Ax'+Bz-'+Gc+D, 

in.  which  A,  B,  C,  D,  E  may  be  positive,  negative,  or  evanescent, 
excepting  those  cases  in  which  the  equation  would  thus  be- 
come one  of  an  inferior  order  of  curves. 

He  distinguished  sixty-five  different  species  of  curves  com- 
prehended under  the  first  equation ;  four  new  species  were 
subsequently  discovered  by  Sterling,  and  four  mo  e  by  De 
Gua. 

The  second  equation  comprehends  only  one  species  of 
curves,  to  which  Newton  has  given  the  name  of  Trident. 

The  third  equation  includes  five  species,  each  possessing  two 
parabolic  branches ;  among  these  is  the  semi-cubical  parabola. 

The  fourth  equation  comprehends  only  one  species  of  curves, 
commonly  called  the  cubical  parabola. 

There  are,  therefore,  eighty  different  species  of  lines  of  the 
third  order. 

(135.)  Lines  of  the  fourth  order  have  their  equations  of  the 
form 

At/4 + By  3x + Cy*x°-  +  Byx3 + Ex* ' 
+  Fy~  +Gy2x  +Hyx*  +  Kx 

i-Lif    +Myx  +Nca  J» 
+  Py    +Qx 
+R 


Classification   of   Algebraic   Curves.       105 

Lines  of  the  fourth  order  are  divided  by  Euler  into  146 
classes,  and  these  comprise  more  than  5000  species. 

As  to  the  fifth  and  higher  orders  of  lines,  their  number  has 
precluded  any  attempt  to  arrange  them  in  classes. 

(136.)  A  family  of  curves  is  an  assemblage  of  several  curves 
of  different  kinds,  all  defined  by  the  same  equation  of  an  inde- 
terminate degree.  Thus,  every  curve  whose  abscissas  are 
proportional  to  any  power  of  the  ordinates  is  called  a  parabola. 
Hence  the  number  of  parabolas  is  indefinite.  Of  these  some 
of  the  most  remarkable  have  received  specific  names.  The 
common  parabola  is  sometimes  called  the  quadratic  parabola, 

and  its  equation  is  of  the  form \f=ax. 

The  equation  of  the  cubical  parabola  is y3=ax. 

The  equation  of  the  biquadratical  parabola  is  .  .  .  yi=ax, 
etc.  etc.  etc. 

The  equation  of  the  semi-cubical  parabola  is    .     .         y*=ax 

The  equation  of  the  semi-biquadratical  parabola  is        y~s=ax 

etc.  etc.  etc. 

All  of  these  parabolas  are  included  it  the  equation     .    y,=ax 


SECTION    IX. 


TRANSCENDENTAL  CURVES. 


(137.)  Curves  may  be  divided  into  two  general  classes, 
algebraic  and  transcendental. 

When  the  relation  between  the  ordinate  and  abscissa  of  a 
curve  can  be  expressed  entirely  in  algebraic  terms,  it  is  called 
an  algebraic  curve ;  when  this  relation  can  not  be  expressed 
without  the  aid  of  transcendental  quantities,  it  is  called  a  trans- 
cendental curve. 

Among  transcendental  curves,  the  cycloid  and  the  logarithmic 
curve  are  the  most  important.  The  logarithmic  curve  is  use- 
ful in  exhibiting  the  law  of  the  diminution  of  the  density  of  the 
atmosphere ;  and  the  cycloid  in  investigating  the  laws  of  the 
pendulum,  and  the  descent  of  heavy  bodies  toward  the  center 
of  the  earth. 

The  spirals  have  many  curious  properties,  and  are  employ- 
ed in  the  volutes  of  the  Ionic  order  of  architecture. 


CYCLOID. 


(138.)  A  cycloid  is  the  curve  described  by  a  point  in  the  cir- 
cumference of  a  circle  rolling  in  a  straight  line  on  a  plane. 


AND 

Thus,  if  the  circle  EPN  be  rolled  along  a  straight  line  AC, 
any  point  P  of  the  circumference  will  describe  a  curve  which 
is  called  the  cycloid.  The  circle  EPN  is  called  the  generating 
circle,  and  P  the  generating  point. 

When  the  point  P  has  arrived  at  C,  having  described  the  arc 
ABC,  if  it  continue  to  move  on,  it  will  describe  a  second  arc 
similar  to  the  first,  a  third  arc,  and  so  on,  indefinitely.  As, 
however,  in  each  revolution  of  the  generating  circle,  an  equal 


Logarithmic   Curve. 


107 


curve  is  described,  it  is  only  necessary  to  examine  the  curve 
ABC  described  in  one  revolution  of  the  generating  circle. 

(139.)  After  the  circle  has  made  one  revolution,  every  point 
of  the  circumference  will  have  been  in  contact  with  AC,  and 
the  generating  point  will  have  arrived  at  C.  The  line  AC  will 
be  equal  to  the'  circumference  of  the  generating  circle,  and  is 
called  the  base  of  the  cycloid.  The  line  BD,  drawn  perpen- 
dicular to  the  base  at  its  middle  point,  is  called  the  axis  of  the 
cycloid,  and  is  equal  to  the  diameter  of  the  generating  circle. 

Proposition  I. — Theorem. 

(140.)   The  equation  of  the  cycloid  is 

x  =  arc  whose  versed  sine  is  y  —  V2ry—y\ 
where  r  represents  the  radius  of  the  generating  circle. 

Let  us  assume  the 
point  A  as  the  origin 
of  co-ordinates,  and 
let  us  suppose  that 
the  generating  point 
has  described  the  arc 
AP.  If  N  designate  ^  R 
the  point  at  which  the  generating  circle  touches  the  base,  it  is 
plain  that  the  line  AN  will  be  equal  to  the  arc  PN.  Through 
N  draw  the  diameter  EN,  which  will  be  perpendicular  to  the 
base.  Through  P  draw  PH  parallel  to  the  base,  and  PR  per- 
pendicular to  it.  Then  PR  will  be  equal  to  HN,  which  is  the 
versed  sine  of  the  arc  PN. 

Let  us  put  EN=2r,  AR=x,  and  PR  or  HN=?/;  we  shall 

then  have,  by  Geom.,  Prop.  XXII.,  Cor.,  B.  IV., 

RN=PH=  VHNXHE=  Vy(2r-ij)=V2ry-y\ 
and  AR=AN-RN=arc  PN-PH. 

Also,       PN  is  the  arc  whose  versed  sine  is  HN  or  y. 
Substituting  the  values  of  AR,  AN,  and  RN,  we  have 
re = arc  whose  versed  sine  is  y—  <</2ry—y'i, 
which  is  the  equation  of  the  cycloid. 

LOGARITHMIC  CURVE. 
(141.)  The  logarithmic  curve  takes  its  name  from  the  prop- 
erty that,  when  referred  to  rectangular  axes,  any  abscissa  is 


108 


Analytical  Geometry. 


equal  to  the  logarithm  of  the  corresponding  ordinate.     The 
equation  of  the  curve  is,  therefore, 

x=\og.  y. 

If  a  represent  the  base  of  a  system  of  logarithms,  we  shall 
have  (Algebra,  Art.  335), 

ax=y. 

(142.)  From  this  equation,  we  can  easily  describe  the  curve 
by  points.  Let  the  line  AB 
be  taken  for  unity ;  and  let 
AC  be  divided  into  portions, 
each  equal  to  AB.  Let  a,  the 
base  of  the  system  of  loga- 
rithms, be  taken  equal  to  any 
assumed  value,  as  1.6,  and  let 
a2,a3,etc,  correspond  in  length 
with  the  different  powers  of  a. 
Then  the  distances  from  A  to 
1,  2,  3,  etc.,  will  represent  the  logarithms  of  a,  a2,  a3,  etc. 

The  logarithms  of  numbers  less  than  a  unit  are  negative,  and 
these  are  represented  by  portions  of  the  line  AD  to  the  left  of 
the  origin. 

(143.)  If  the  curve  be  continued  ever  so  far  to  the  left  of  A, 
it  will  never  meet  the  axis  of  abscissas.  The  ordinates  di- 
minish more  and  more,  but  can  never  reduce  to  zero,  while  x 
is  a  finite  quantity.  When  the  ordinate  becomes  infinitely 
small,  the  abscissa  becomes  infinitely  great  and  negative. 
This  corresponds  with  Algebra  (Art.  337),  where  it  is  shown 
that  the  logarithm  of  zero  is  infinite  and  negative. 

(144.)  We  may  construct  the  curve  for  any  system  of  loga- 
rithms in  a  similar  manner.     Thus,  for  the  Naperian  system, 

a    =  2.718, 

a2    =   7.389, 

a3   =20.085, 

a~'=  0.368, 

a-3=  0.135. 

If  we  erect  at  the  point  A  an 

ordinate  equal  to  unity ;  at  the 

point    1    an   ordinate   equal   to 

2.718 ;  at  the  point  2  an  ordi- 


Spiral   of   Archimedes. 


100 


nate  equal  to  7.389,  etc. ;  at  the  point  -1  an  ordinate  equal  to 
0.368,  etc.,  the  curve  passing  through  the  extremities  of  these 
ordinates  will  be  the  logarithmic  curve  for  the  Naperian  base. 

SPIRALS. 
(145.)  A  spiral  is  a  curve  described  by  a  point  which  moves 
along  a  right  line  in  accordance  with  some  prescribed  law,  the 
line  having  at  the  same  time  a  uniform  angular  motion. 

Thus,  let  PD  be  a  straight 
line  which  revolves  uniformly 
around  the  point  P;  and,  at 
the  same  time,  let  a  point  move 
from  P  along  the  linePD,arriv- 
'  ing  successively  at  the  points 
A,  B,  C,  etc.,  it  will  trace  out 
a  curve  called  a  spiral. 

(146.)  The  fixed  point  P 
about  which  the  right  line 
moves,  is  called  the  pole  of  the 

spiral.  The  portion  of  the  spiral  generated  while  the  line 
makes  one  revolution,  is  called  a  spire ;  and  if  the  revolutions 
of  the  radius  vector  are  continued,  the  generating  point  will 
describe  an  indefinite  spiral,  and  any  straight  line  drawn 
through  the  pole  of  the  spiral,  will 
intersect  it  in  an  infinite  number 
of  points. 

With  P  as  a  center,  and  PA  as 
a  radius,  describe  the  circumfer- 
ence ADE ;  the  angular  motion  of 
the  radius  vector  about  the  pole 
may  be  measured  by  the  arcs  of 
this  circle  estimated  from  A. 

SPIRAL  OF  ARCHIMEDES. 
(147.)  While  the  line  PD  revolves  uniformly  about  P,  let 
the  generating  point  move  uniformly,  also,  along  the  line  PD,  it 
will  describe  the  spiral  of  Archimedes. 

Proposition  II. — Theorem. 
(148.)   The  equation  of  the  spiral  of  Archimedes  is 


110 


Analytical  Geometry. 


__£_ 

where  r  represents  the  radius  vector,  and  t  the  measuring  aic 
estimated  from  A. 

For,  from  the  definition,  the  radius  vectors  are  proportional 
to  the  measuring  arcs  estimated  from  A  ;  that  is, 

PM  :  PD  : :  arc  AD  :  circ.  ADE. 

Designate  the  radius  vector  PM  by  r,  PA  by  a,  and  the 
measuring  arc  estimated  from  A  by  t;  then  we  shall  have 

r  :  a  : :  t  :  2na. 

at        t 
Whence  r=- — =— . 

2na     2n 

(149.)  This  spiral  may  be  con- 
structed as  follows :  divide  a  cir- 
cumference into  any  number  of 
equal  parts,  as,  for  example,  eight ; 
and  the  radius  AP  into  the  same 
number  of  equal  parts.  On  the 
radius  PB  lay  off  one  of  these 
parts ;  on  PC  two  ;  on  PD  three, 
etc.  The  curve  passing  through 
these  points  will  be  the  spiral  of 
Archimedes,  for  the  radius  vectors  are  proportional  to  the  arcs 
AB,  AC,  etc.,  of  the  measuring  circle. 


HYPERBOLIC  SPIRAL. 


(150.)  While  the  line 
PB  revolves  uniformly 
about  P,  let  the  genera- 
ing  point  move  along 
the  line  PB  in  such  a 
manner,  that  the  radius  Df 
vectors  shall  be  inverse- 
ly proportional  to  the 
corresponding  arcs,  it 
will  describe  the  hyper- 
bolic spiral. 


E" 


Logarithmic   Spiral. 


Ill 


Proposition  III. — Theorem. 
(151.)   The  equation  of  the  hyperbolic  spiral  is 

a 

where  r  represents  the  radius  vector,  t  the  measuring  aic,  and 
a  is  a  constant  quantity. 
For,  from  the  definition, 

PB  :  PM  : :  circ.  ABDE  :  arc  AB. 
Let  us  designate  the  radius  vector  by  r,  and  the  measuring 
arc  estimated  from  A  by  t,  calling  PM  unity,  we  shall  have 
r  :  1  ::2tt  :t. 
2n 
Whence  r=~T ' 

or,  representing  the  constant  2n  by  a,  we  have 

a 
r=-,  or  at~  . 

(152.)  Scholium.  The  two  preceding  spirals  are  included  in 
the  general  equation 

r=af, 
where  n  may  be  either  positive  or  negative. 

(153.)  The  hyperbolic  spiral  may  be  constructed  as  follows: 
Describe  a  circle  ACDF, 
and  divide  its  circumfer- 
ence into  any  number  of 
equal  parts,  AB,  BC,  CD, 
etc.  Then  take  PG  equal 
to  one  half  of  PB,  PH  T>\ 
equal  to  one  third  of  PB, 
PI  equal  to  one  fourth  of 
PB,  etc.,  the  curve  pass- 
ing through  the  points  B, 
G,  H,  I,  etc.,  will  be  a  hy- 
perbolic spiral,  because  the  radius  vectors  are  inversely  pro- 
portional to  the  correspoiding  arcs  estimated  from  A. 

LOGARITHMIC  SPIRAL. 
(154.)  While  the  line  PA  revolves  uniformly  about  P,  let 
the  geneiating  point  move  along  PA  in  such  a  manner  thai 


112  Analytical   Geometry. 

the  logarithm  of  the  radius  vector  may  be  proportional  to  the 
measuring  arcs,  it  will  describe  the  logarithmic  spiral. 

Proposition  IV. — Theorem. 
(155.)   The  equation  of  the  logarithmic  spiral  is 
t  =  a  log.  r, 
where  r  represents  the  radius  vector,  and  t  the  measuring  arc. 
For  this  equation  is  but  an  expression  of  the  definition. 
(156.)  The  logarithmic  spiral  may  be  constructed  as  follows  : 
Divide  the  arc  of  a  circle  ACE 
into  any  number  of  equal  parts, 

AB,  BC,  CD,  etc.,  arid  upon  the 
radii  drawn  to  the  points  of  divi- 
sion, take  PL,PM,  PN,  etc.,  in  geo- 
metrical progression.  The  curve 
passing  through  the  points  L,  M, 
N,  etc.,  will  be  the  logarithmic 
spiral ;  for  it  is  evident  that  AB, 

AC,  etc.,  being  in  arithmetical  progression,  are  as  the  loga- 
rithms of  PL,  PM,  etc.,  which  are  in  geometrical  progression. 
See  Algebra,  Art.  315. 


DIFFERENTIAL   CALCULUS. 


SECTION    I. 

DEFINITIONS  AND  FIRST  PRINCIPLES-DIFFERENTIATION  OF 
ALGEBRAIC  FUNCTIONS. 

Article  (157.)  In  the  Differential  Calculus,  as  in  Analytical 
Geometry,  there  are  two  classes  of  quantities  to  be  considered, 
viz.,  variables  and  constants. 

Variable  quantities  are  generally  represented  by  the  last  let- 
ters of  the  alphabet,  x,  y,  z,  etc.,  and  any  values  may  be  assign- 
ed to  them  which  will  satisfy  the  equations  into  which  they 

enter. 

Constant  quantities  are  generally  represented  by  the  first 
letters  of  the  alphabet,  a,  b,  c,  etc.,  and  these  always  retain  the 
same  values  throughout  the  same  investigation. 

Thus,  in  the  equation  of  a  straight  line, 
y=ax  +  b, 
the  quantities  a  and  b  have  but  one  value  for  the  same  line, 
while  x  and  y  vary  in  value  for  every  point  of  the  line. 

(158.)  One  variable  is  said  to  be  a  function  of  another  varia- 
ble, when  the  first  is  equal  to  a  certain  algebraic  expression 
containing  the  second.     Thus,  in  the  equation  of  a  straight  line 

y=ax-\-b, 
y  is  a  function  of  x. 

So,  also,  in  the  equation  of  a  circle, 

and  in  the  equation  of  the  ellipse, 

B 


y=—V'2Ax  —  x\ 

(159.)  When  we  wish  merely  to  denote  that  y  is  dependent 
upon  x  for  its  value,  without  giving  the  particular  expression 
which  shows  the  value  of  x,  we  employ  the  notation 

H 


114  Differential   Calculus. 

y=F(x),  or  y=f(x), 
or  x=F(y)fovx=f(y), 

which  expressions  are  read,  y  is  a  function  of  a;,  or  x  is  a  func- 
tion of  y. 

To  denote  a  function  containing  two  independent  variables, 
as  x  and  y,  we  inclose  the  variables  in  a  parenthesis,  and  place 
the  sign  of  function  before  them.     Thus,  the  equation 

u=ay+bx'2 
may  be  expressed  generally  by 

u=f{x,  y), 
which  is  read,  u  is  a  function  of  x  and  y,  and  simply  shows 
that  u  is  dependent  for  its  value  upon  both  x  and  y. 

(160.)  An  explicit  or  expressed  function  is  one  in  which  the 
value  of  the  function  is  directly  expressed  in  terms  of  the  varia 
ble  and  constants,  as  in  the  equation 

y=axi.  +  b. 
An  implicit  or  implied  function  is  one  in  which  the  value  of 
the  function  is  not  directly  expressed  in  terms  of  the  variable 
and  constants,  as  in  the  equation 

ys— 3ayx-\-xs=0, 
where  the  form  of  the  function  that  y  is  of  x  can  be  ascertain 
ed  only  by  solving  the  equation. 

(161.)  An  increasing  function  is  one  which  is  increased 
when  the  variable  is  increased,  or  decreased  when  the  varia- 
ble is  decreased.     Thus,  in  the  equation  of  a  straight  line, 

y—ax+b, 
if  the  value  of  a;  is  increased,  the  value  of  y  will  also  increase; 
or,  if  x  is  diminished,  the  value  of  y  will  diminish. 

A  decreasing  function  is  one  which  is  decreased  when  the 
variable  is  increased,  and  increased  when  the  variable  is  de- 
creased.    Thus,  in  the  equation  of  the  circle, 

?/=  VR'-x% 
the  value  of  y  increases  when  x  is  diminished,  and  decreases 
when  x  is  increased. 

In  the  equation  y=  VIV— x\  x  is  called  the  independent 
variable,  and  y  the  dependent  variable,  because  arbitrary  values 
are  supposed  to  be  assigned  to.r,  and  the  corresponding  values 
of  y  are  deduced  from  the  equation. 


Definitions    and   first   Principles. 


115 


(1G2.)  The  limit  of  a  variable  quantity  is  that  value  which 
it  continually  approaches,  so  as,  at  last,  to  differ  from  it  by 
less  than  any  assignable  quantity. 

Thus,  if  we  have  a  regular  polygon  inscribed  in  a  circle, 
and  if  we  inscribe  another  polygon  having  twice  the  number  of 
sides,  the  area  of  the  second  will  come  nearer  to  the  area  of  the 
circle  than  that  of  the  first.  By  continuing  to  double  the  num- 
ber of  sides,  the  area  of  the  polygon  will  approach  nearer  and 
nearer  to  that  of  the  circle,  and  may  be  made  to  differ  from  it 
by  a  quantity  less  than  any  finite  quantity.  Hence  the  circle 
is  said  to  be  the  limit  of  all  its  inscribed  polygons. 

So,  also,  in  the  equation  of  a  circle, 

the  value  of  y  increases  as  the  point  P  ad- 
vances from  A  to  B,  at  which  point  it  be- 
comes equal  to  the  radius  of  the  circle. 
As  the  point  P  advances  from  B  to  C,  the    J 
value  of  y  diminishes  until  at  C  it  is  re- 
duced to  zero.     The  radius  of  the  circle 
is,  therefore,  the  limit  which  the  value  of 
y  can  never  exceed.     So,  also,  in  the  same  equation,  the  radius 
of  the  circle  is  the  limit  which  the  value  of  a;  can  never  exceed. 
If  we  convert  }  into  a  decimal  fraction,  it  becomes 
.1111,  etc., 
or  -'-  +  -!-  + ' 4- i 4-   etr 

Hence  the  sum  of  the  terms  of  this  series  approaches  to  the 
value  of  i,  but  can  never  equal  it  while  the  number  of  terms  is 
finite.  The  limit  of  the  sum  of  the  terms  of  this  series  is  there- 
fore i. 

So,  also,  the  sum  of  the  series, 

1+i+i+i+rV.  etc., 
approaches  nearer  to  2,  the  greater  the  number  of  terms  we 
employ  ;  and,  by  taking  a  sufficient  number  of  terms,  the  sum 
of  the  series  may  be  made  to  differ  from  2  by  less  than  any 
quantity  we  may  please  to  assign.  The  limit  of  the  sum  of 
the  terms  of  this  series  is  therefore  2. 

(1G3.)  When  two  magnitudes  decrease  simultaneously,  they 
may  approach  continually  toward  a  ratio  of  equality,  or  to- 
ward some  other  definite  ratio.     Thus,  let  a  point  P  be  sup- 


1 1G  Differential   Calculus. 

posed  to  move  on  the  circumference  of  a  circle  toward  a  fixed 
point  A.     The  arc  AP  will  diminish   at  r> 

the  same  time  with  the  chord  AP,  and,  by 
bringing  the  point  P  sufficiently  near  to 
A,  we  may  obtain  an  arc  and  its  chord, 
each  of  which  shall  be  smaller  than  any 
given  line,  and  the  arc  and  the  chord  con- 
tinually approach  to  a  ratio  of  equality. 

But  the  ratio  of  two  magnitudes  does  not  necessarily  ap 
proach  to  equality,  because  the  magnitudes  are  indefinitely 
diminished.     Thus,  take  the  two  series, 

*»      3'      6»      10'      I  5»      21'      2B>    Llv"! 

1      I     1    •_»_     _"_     JL     JL    pic 

X>      4'      9'      1  6>      2  5'      36'     4  9>ClU 

The  ratio  of  the  corresponding  terms  is, 

1        4        9        16        25        36        49      pfn 
x>      3'     6'      10'      15»      21'      2  8'    Ctl" 

The  ratio  here  increases  at  every  step,  but  not  without  limit 
However  far  the  two  series  may  be  continued,  the  ratio  of  the 
corresponding  terms  is  never  so  great  as  2,  though  it  may  be 
made  to  differ  from  2  by  less  than  any  assignable  quantity. 
The  limit  of  the  ratio  of  the  corresponding  terms  of  the  two 
series  is  therefore  2. 

(164.)  If  a  variable  quantity  increase  uniformly,  then  other 
quantities,  depending  on  this  and  constant  quantities,  may 
either  vary  uniformly,  or  according  to  any  law  whatever. 

Thus,  in  the  equation  of  a  straight  line, 

y=2x-\-3, 
if  we  make  x=l,  we  find  y=5, 

x=2,      "        y=l, 
x=3,       "         y=9, 
etc.  etc. ; 

that  is,  when  x  increases  uniformly,  y  increases  uniformly. 
Again,  take  the  equation  of  the  parabola, 

y—  V4x, 

if  we  make  x=l,  we  find  y=2.000, 

x=2,       "        y=2.828, 

z=3,       "         y=3AQ4, 

x=4,       "         y=4.000, 

etc.  etc., 


\ 

Differentiation    of   Algebraic   Functions.  117 

where,  although  x  increases  uniformly,  y  does  not  increase 
uniformly. 

(165.)  If  the  side  of  a  square  increases  uniformly,  the  area 
does  not  increase  uniformly.  Thus,  let  AB 
be  the  side  of  a  square,  and  let  it  increase 
uniformly  by  the  additions  Ba,  ab,  be,  etc.,  DF 
and  let  squares  be  described  on  these  lines, 
as  in  the  annexed  figure  ;  then  it  is  obvious 
that  the  square  on  the  side  Aa  exceeds  that 
described  on  the  side  AB,  by  twice  the  rect-  A  Bab  c 

angle  ABxBa,  together  with  the  square  on  Ba.  The  square 
described  on  Ab  has  received  a  further  increment  of  two  equal 
rectangles,  together  with  three  times  the  square  on  Ba ;  the 
square  on  AC  has  received  a  further  increment  of  two  equal 
rectangles  and  five  times  the  square  on  Ba.  Hence,  when  the 
side  of  the  square  varies  uniformly,  the  area  does  not  vary 
uniformly. 

Thus,  suppose  the  side  of  a  square  is  equal  to  10  inches,  and 
let  it  increase  uniformly  one  inch  per  minute,  so  as  to  become 
successively  11,  12,  etc.,  inches. 

While  the  side  increases  from  10  to  11  inches,  the  area  in- 
creases from  100  to  121  inches=21  inches. 

While  the  side  increases  from  11  to  12  inches,  the  area  in- 
creases from  121  to  144  inches=23  inches. 

While  the  side  increases  from  12  to  13  inches,  the  area  in- 
creases from  144  to  109  inches=25  inches. 

etc.,  etc.,  etc. 

Hence  the  rate  of  increase  of  the  area  depends  upon  the 
length  of  the  side.  When  the  side  is  11  inches,  the  area  is  in- 
creasing more  rapidly  than  when  the  side  was  10  inches. 

(1G6.)  There  is  an  important  distinction  between  the  abso- 
>ute  increase  of  a  variable  quantity,  and  its  rate  of  increase. 
By  the  rate  of  increase  at  any  instant  we  understand  what 
would  have  been  the  absolute  increase  if  this  increase  had  been 
uniform.  Thus,  while  the  side  of  a  square  increases  from 
11  to  12  inches  in  one  minute,  the  area  increases  from  121  to 
144  inches.  The  absolute  increase  of  the  area  is  23  inches; 
but  the  rate  of  increase  of  the  area  when  '.he  side  was  11 
inches  was  such  as  would  have  given  an  increase  of  less  than 
23  inches  per  minute  ;  and  when  the  side  was  12  inches  the 


118  Differential   Calculus. 

rate  of  increase  was  such  as  would  have  given  an  increase?  or 
more  than  23  inches  per  minute. 

While,  therefore,  the  rate  of  increase  of  the  side  of  a  square 
is  uniform,  the  rate  of  increase  of  its  area  is  continually 
changing. 

The  object  of  the  Differential  Calculus  is  to  determine  the 
ratio  between  the  rate  of  variation  of  the  independent  variable 
and  that  of  the  function  into  which  it  enters. 

Proposition  I. — Theorem. 

(167.)  The  rate  of  variation  of  the  side  of  a  square  is  to  thai 
of  its  area,  in  the  ratio  of  unity  to  twice  the  side  of  the  square. 

If  the  side  of  a  square  be  represented  by  x,  its  area  will  be 
represented  by  x*.  When  the  side  of  the  square  is  increased 
by  h  and  becomes  x+h,  the  area  will  become  (x-t-h)9,  which 
is  equal  to 

x'+2xh+h\ 

While  the  side  has  increased  by  //,  the  area  has  increased  by 
2xh-{h\  If,  then,  we  employ  h  to  denote  the  rate  at  which  x 
increases,  2xh-\-h*  would  have  denoted  the  rate  at  which  the 
area  increased  had  that  rate  been  uniform  ;  in  which  case  we 
should  have  had  the  following  proportion  : 

rateof  increase  of  the  side :  rate  of 'increase  of the  area  ::h:  2x11+1? 
or  as  l :  2x  +h. 

But  since  the  area  of  the  square  increases  each  instant  more 
and  more  rapidly,  the  quantity  2x-\-h  is  greater  than  the  incre- 
ment which  would  have  resulted  had  the  rate  at  which  the 
square  was  increasing  when  its  side  became  x  continued  uni- 
form ;  and  the  smaller  h  is  supposed  to  be,  the  nearer  does  the 
increment  2x + h  approach  to  that  which  would  have  result- 
ed had  the  rate  at  which  the  square  was  increasing  when  its 
side  became  x  continued  uniform.  When  h  is  equal  to  zero, 
this  ratio  becomes  that  of 

1  to  2x, 
which  is,  therefore,  the  ratio  of  the  rate  of  increase  of  the  side 
tc  that  of  the  area  of  the  square,  when  the  side  is  equal  to  x. 

(168.)  Illustration.  If  the  side  of  a  square  be  10  feet,  its 
area  will  be  100  feet      If  the  side  be  mcreased  to  11  feet,  its 


Differentiation   of   Algebraic   Functions.  119 

area  will  become  121  feet ;  the  area  has  increased  21  feet ; 
and  the  ratio  of  the  increment  of  the  side  to  that  of  the  area  is 
as  1  to  21. 

When  the  length  of  the  side  was  10.1  feet,  its  area  was 
102.01  ;  the  area  had  increased  2.01  feet ;  and  the  ratio  of  the 
increment  of  the  side  to  that  of  the  area,  was  as  0.1  to  2.01,  or 
1  to  20.1. 

When  the  length  of  the  side  was  10.01  feet,  its  area  was 
100.2001  ;  and  the  ratio  of  the  increment  of  the  side  to  that  of 
the  area,  was  as  1  to  20.01. 

When  the  length  of  the  side  was  10.001  feet,  the  ratio  was 
as  1  to  20.001. 

Hence  we  see  that  the  smaller  is  the  increment  of  the  side  of 
the  square,  the  nearer  does  the  ratio  of  the  increments  of  the 
side  and  area  approach  to  the  ratio  of  1  to  20.  This,  there- 
fore, was  the  ratio  of  the  rates  of  increase  at  the  instant  the 
side  was  equal  to  10  feet ;  and  this  ratio  is  that  of  one  to  twice 
ten,  or  twice  the  side  of  the  square. 

We  have  here  another  illustration  of  the  principle  of  Art. 
163,  that  two  magnitudes  which  decrease  simultaneously  may 
continually  approach  toward  some  finite  ratio.  However 
small  we  suppose  the  increment  of  the  side  of  the  square  or 
the  increment  of  the  area  to  become,  the  ratio  of  the  two  in- 
crements continually  approaches  to  that  of  1  to  2x. 

Proposition  II. — Theorem. 

(169.)  The  rate  of  variation  of  the  edge  of  a  cube  is  to  that 
of  its  solidity,  in  the  ratio  of  unity  to  three  times  the  square  of 
the  edge. 

If  the  edge  of  a  cube  be  represented  by  x,  and  its  solidity  by 
u,  then 

u— x3. 
If  the  edge  of  the  cube  be  increased  by  h  so  as  to  become 
z+h,  and  the  corresponding  solidity  be  represented  by  u',  then 
we  shall  have 

u'=  (x + h)  *=z3+ 3x*h + 3xh* + h\ 
The  increment  of  the  cube  is 

u'  -u=3x'h+Sxh',+h\ 
Hence,  if  the  solidity  of  the  cube  had  increased  uniformly  when 


120  Differential   Calculus. 

the  edge  increased  uniformly,  we  should  have  had  the  propoi 
tion, 

rate  of  increase  of  the  edge  :  rate  of  increase  of  the  solidity 

::h:  3x'2h  +  3xh2+h\ 
or  as  1  :  3x"   +  3xh  +h\ 

But  since  the  solidity  at  each  instant  increases  more  and 
more  rapidly,  the  increment  3x2  +  3xh  +  If  is  greater  than  that 
which  would  have  resulted  had  the  rate  of  increase  when  its 
edge  became  x  continued  uniform.  Now  the  smaller  h  be- 
comes, the  nearer  does  the  increment  3x^+3x11+)^  approach 
to  that  which  would  have  resulted  had  the  rate  at  which  the 
cube  was  increasing  when  its  edge  became  x  continued  uni- 
form.    When  h  is  equal  to  zero,  this  ratio  becomes  that  of 

1  to  3x", 
which  is,  therefore,  the  ratio  of  the  rate  of  increase  of  the  edge 
to  that  of  the  solidity,  when  the  edge  is  equal  to  x. 

(170.)  The  rate  of  variation  of  a  function  or  of  any  variable 
quantity  is  called  its  differential,  and  is  denoted  by  the  lettei 
d  placed  before  it.     Thus,  if 

u=x3, 
then  dx  :  du  : :  1  :  3x*. 

The  expressions  dx,  du  are  read  differential  of  a:,  differential 
of  u,  and  denote  the  rates  of  variation  of  a;  and  u. 

If  we  multiply  together  the  extremes  and  the  means  of  the 
preceding  proportion,  we  have 

du=3x'idx, 

which  signifies  that  the  rate  of  increase  of  the  function  u  is 
3x'  times  that  of  the  variable  x. 

If  we  divide  each  member  of  the  last  equation  by  dx,  we 
have 

du 
dx 
which  expresses  the  ratio  of  the  rate  of  variation  of  the  func- 
tion to  that  of  the  independent  variable,  and  is  called  the  dif- 
ferential coefficient  of  u  regarded  as  a  function  of  x. 

(171.)  Illustration.  If  the  edge  of  a  cube  be  10  feet,  its 
solidity  will  be  1000  feet.  If  the  edge  be  increased  to  11  feet, 
its  aoliiity  will  be  1331   feet;  the  solidity  has  increased  331 


Differentiation    of    Algebraic   Functions.  121 

feet ;  and  the  ratio  of  the  increment  of  the  edge  to  that  of  the 
solidity  is  as  1  to  331. 

When  the  length  of  the  edge  was  10.1  feet,  its  solidity  was 
1030.301  ;  the  solidity  had  increased  30.301  feet;  and  the  ratio 
of  the  increment  of  the  edge  to  that  of  the  solidity  was  as  0.1 
to  30.301,  or  1  to  303.01. 

When  the  length  of  the  edge  was  10.01  feet,  its  solidity  wag 
"003.003001  ;  and  the  ratio  of  the  increment  of  the  edge  to 
hat  of  the  solidity  was  as  1  to  300.3001. 

When  the  length  of  the  edge  was  10.001  feet,  the  ratio  was 
as  1  to  300.030001. 

Hence  we  see  that  the  smaller  is  the  increment  of  the  edge 
of  the  cube,  the  nearer  does  the  ratio  of  the  increments  of  the 
edge  and  solidity  approach  to  the  ratio  of  1  to  300.  This, 
therefore,  was  the  ratio  of  the  rates  of  increase  at  the  instant 
the  edge  was  equal  to  10  feet ;  and  this  ratio  is  that  of  one  to 
three  times  the  square  often. 

(172.)  It  will  be  seen  from  these  examples  that,  in  order  to 
discover  the  rate  of  variation  of  a  function,  we  ascribe  a  small 
increment  to  the  independent  variable,  and  learn  the  corre- 
sponding increment  of  the  function.  We  then  observe  toward 
what  limit  the  ratio  of  these  increments  approaches,  as  the  in- 
crement of  the  variable  is  diminished,  which  limit  can  only  be 
attained  when  the  increment  of  the  variable  is  supposed  to  be- 
come zero.  This  limit  expresses  the  ratio  of  the  rates  of 
variation  of  the  function  and  the  independent  variables,  at  the 
instant  when  the  variable  was  equal  to  x. 

But  because,  in  order  to  find  the  value  of  the  differential  co- 
efficient, we  make  h  equal  to  zero,  it  must  not  be  inferred  that 
dx  and  du  are  therefore  equal  to  zero,  du  denotes  the  rate  of 
variation  of  the  function  u,  and  dx  the  rate  of  variation  of  the 
variable  x;  and  since  only  their  ratio  is  determined,  either  of 
them  may  have  any  value  whatever,  dx  may,  there 'ore,  be 
supposed  to  have  a  very  small  or  a  very  large  value  at  r  leasure. 

Proposition  III. — Theorem. 

(173.)   The  differential  coefficient  of  the  function 

u=x* 
is  4x\ 

If  we  suppose  x  to  be  increased  by  any  quantity  h,  and  des 


J 22  Differential   Calculus. 

ignate  by  u'  the  new  value  of  the  function  under  this  supposi 
tion,  we  shall  have 

u'=(x+hy; 

or,  expanding  the  second  member  of  the  equation,  we  have 

u'=xi+4xah+6xVf+4xhs+h\ 

If  we  subtract  from  this  the  original  equation,  we  obtain 

u'-u= 4x*h  +  Gx  7i2 + 4xh3 + h*. 

Hence  we  see  that  if  the  variable  x  is  increased  by  h,  the  func- 
tion u  will  be  increased  by 

4x9h+6x*h*+4xha+hl. 

If  both  members  of  the  last  equation  be  divided  by  h,  we 
shall  have 

■■4x*+6xVi+4xh*+h3, 


h 

which  expresses  the  ratio  of  the  increment  of  the  function  u  to 
that  of  the  variable  x.  The  first  term  4x3  of  this  ratio  is  in- 
dependent of  h,  so  that,  however  we  vary  the  value  of  h,  this 
first  term  will  remain  unchanged,  but  the  subsequent  terms  are 
dependent  on  h. 

If  we  suppose  h  to  diminish  continually,  the  value  of  this 
ratio  will  approach  to  that  of  4x3,  to  which  it  will  become 
equal  when  h  equals  zero.     This,  therefore,  is  the  ratio  of  the 
rate  of  increase  of  the  independent  variable  to  that  of  the  func 
tion.  at  the  instant  the  variable  was  equal  to.r.    Hence,  Art.  172, 

du 

— =4x  . 

dx 

(174.)  The  method  here  exemplified  is  applicable  to  the  de- 
termination of  the  differential  coefficient  of  any  function  of  a 
single  variable,  and  is  expressed  in  the  following 

Rule. 

Give  to  the  variable  any  arbitrary  increment  h,  and  find  the 
corresponding  value  of  the  function ;  from  which  subtract  its 
primitive  value.  Divide  the  remainder  by  the  increment  h,  and 
find  the  limit  of  this  ratio,  by  making  the  increment  equal  to 
zero  ;  the  result  will  be  the  differential  coefficient. 

Ex.  1.  If  vb  increase  uniformly  at  the  rate  of  2  inches  per 


Differentiation*   of    Algebraic   Functions.  123 

second,  at  what  rate  does  the  value  of  the  expression  2x*  in- 
crease when  x  equals  6  inches  ? 

Ans.  48  inches  per  second. 
Ex.  2.  If  x  increase  uniformly  at  the  rate  of  3  inches  per 
second,  at  what  rate  does  the  value  of  the  expression  4x3  in- 
crease when  x  equals  10  inches  ? 

Ans. 
Ex.  3.  If  x  increase  uniformly  at  the  rate  of  5  inches  per 
second,  at  what  rate  does  the  value  of  the  expression  2x*  in- 
crease when  x  equals  4  inches  ? 

Ans. 

Proposition  IV. — Theorem. 

(175.)  To  obtain  the  differential  of  any  power  of  a  variable, 
we  must  diminish  the  exponent  of  the  power  by  unity,  and  then 
multiply  by  the  primitive  exponent,  and  by  the  differential  of  the 
variable. 

To  prove  this  proposition,  let  us  take  the  function 

u=xn, 
and  suppose  x  to  become  x  +  h,  then 

u'=(x+h)n. 
Developing  the  second  member  of  this  equation  by  the  Bi- 
nomial theorem,  we  have 

u'=xn + 7^.^■n-'^+7^     ~   -ar""3/*3  + ,  etc. 

Subtracting  from  this  the  original  equation,  we  have 

,_     n(n  —  1) 
u'  —  u=nxn ~'/H — --^ xn   h-+,  etc. 

Dividing  both  members  by  h,  we  have 

U'-U  n-l  ,  rc(n-l) 

— : — =nx     -\ xn  -h+,  etc. 

h  2 

which  expresses  the  ratio  of  the  increment  of  the  function  to 
that  of  the  variable. 

If  now  we  make  h  equal  to  zero,  Art.  174,  the  second  term 
of  the  second  member  of  this  equation  reduces  to  zero,  and 
also  all  the  subsequent  'terms  pf  the  development,  since  they 
contain  powers  of  h.     Hence 


24  Differential   Calculus. 

du 

dx-=n?  > 

or  du=nxn~1dx, 

which  conforms  to  the  proposition  abovre  enunciated. 

Proposition  V. — Theorem. 

(17G.)  The  differential  of  the  product  of  a  variable  quantity 
by  a  constant,  is  equal  to  the  constant  multiplied  by  the  differ- 
ential of  the  variable. 

Suppose  we  have  the  function 

u^ax*. 

When  x  becomes  x-\-h,  we  have 

u'=axi+4ax*h  +  6ax'*h'2+,  etc. 

Also,  u'  —  u=4ax*h+6axVf+,  etc. 

tt  u'  —  u 

Hence  — -. — =4ax  +Gax'h-{-,  etc. 

If  now  we  make  h  equal  to  zero,  Art.  174,  all  the  terms  id 

the  second  member  of  this  equation  except  the  first  disappear 

and  we  have 

du 

-j-=4ax  . 
dx 

or  du=4ax?dx ; 

that  is,  the  differential  of  ax*  is  equal  to  the  differential  of  x* 

multiplied  by  a. 

Proposition  VI. — Theorem. 

(177.)  The  differential  of  a  constant  term  is  zero;  hence  d 
constant  quantity  connected  with  a  variable  by  the  sign  plus  or 
minus,  will  disappear  in  differentiation. 

Suppose  we  have  the  function 

u=b+x\ 

When  x  becomes  x+h,  we  have 

u' = b+x* + 4x*h + GxVi2  -f ,  etc., 

and  u'  —  u=4x3h+6x2/f+,  etc. 

u'  —  u 
Hence  — r—  =4x'  +  Gx*h+.  etc. ; 

h 

and,  making  h  equal  to  zero,  Art.  174,  we  have 


Differentiation   of   Algebraic   Functions.  12D 

du 

—  —  Ax 

dx  ' 

or  du=4x*dx, 

where  the  constant  term  b  has  disappeared  in  differentiation. 

Ex.  1.  What  is  the  differential  of  5ax5l 

Ans.   1 5ax*dx. 

Ex.  2.  What  is  the  differential  of  %x°  +  b? 

Ans. 

Ex.  3.  What  is  the  differential  of  3xb1 

Ans. 

Ex.  4.  What  is  the  differential  of  7aV  +  63? 

Ex.  5.  What  is  the  differential  of  4a¥xt— c? 

Ex.  6.  What  is  the  differential  of  3a3cx°-d? 

Proposition  VII. — Theorem. 

(178.)  If  u  represents  any  function  of  x,  and  we  change  x  into 
x-\-h,  the  new  value  of  the  function  will  consist  of  three  parts  : 

\st.   The  primitive  function  u. 

2d.  The  differential  coefficient  of  the  function  multiplied  by 
the  first  power  of  the  increment  h. 

3d.  A  function  of  x  and  h  multiplied  by  the  second  power  of 
the  increment  h. 

We  have  seen  in  the  preceding  Propositions  that  when  u  is 
a  function  of  x,  and  we  change  x  into  x+h,  the  new  value  of 
the  function  consists  of  a  series  of  terms  which  may  be  ar- 
ranged in  the  order  of  the  ascending  powers  of  h,  and  the  de- 
velopment is  of  the  following  form 

u'=A  +  Bh+Ch2+'Dh*+,  etc. 

Now  when  we  suppose  h  equal  to  zero,  the  second  member 
of  this  equation  reduces  to  A,  and  u1  on  this  supposition  be- 
comes u;  hence 

A=u, 

and  the  development  may  be  written 

u'—u+Bh+Cha+Bh9+,  etc., 
or  u'  =  u+Bh+h\C+Dh+,  etc.). 

If  now  we  represent  C  +  D/t-f- ,  etc.,  by  C,  where  C  is  a 
function  both  of  x  and  h,  we  have 

u'^u  +  Bh+C'h*.  (1) 


126  DIFFERENTIAL     CALCULUS. 

Transposing  and  dividing  by  h,  we  find 
u'  —  u 


B+C'A, 

zero 
:=B; 


h 

and  when  we  make  h  equal  to  zero,  we  have 

du 
dx 
that  is,  B  is  the  differential  coefficient  of  the  function. 

We  see  from  equation  (1)  that  u',  the  new  value  of  the  func- 
tion, consists  of  the  primitive  function  u,  plus  the  differential 
coefficient  of  the  function  multiplied  by  h,  plus  a  function  of 
x  and  h  multiplied  by  h\ 

This  new  value  of  the  function  will  be  frequently  referred  to 
hereafter  under  the  form 

u'=u  +  Ah  +  Bh\  (2) 

Proposition  VIII. — Theorem. 

(179.)  The  differential  of  the  sum  or  difference  of  any  num- 
ber of  functions  dependent  on  the  same  variable,  is  equal  to  the 
sum  or  difference  of  their  differentials  taken  separately. 

Let  us  suppose  the  function  u  to  be  composed  of  several 
variable  terms,  as,  for  example, 

u—y+z—  u, 
where  y,  z,  and  u  are  functions  of  a:. 

If  we  change  x  into  x-\-h,  we  shall  have 

u'— u  =  (y  '—y)  +  (z'  —  z)  —  (u'  —  u). 

But  by  the  preceding  Proposition  y'—y  may  be  put  under 
the  form  of  Ah-rBh*. 

So,  also  z'  —  z  may  be  put  under  the  form  of  A'/i  +  B'A', 
and  u'  —  u  may  be  represented  by  A'7i  +  B'7i8;  that  is, 

u'-u=(Ah+Bh*)+(A'h+B'h*)-(A"h+B"h*). 

Dividing  each  member  by  h,  we  have 

^p=(A+BA)+(A'+B70-(A"+B"A); 

and  making  h  equal  to  zero,  Art.  174,  we  have 

du 

-5-=A+A'-A", 
dx 

or  du=Adx+A'dx  —  A"dx. 


Differentiation   of   Algebraic   Functions.  127 

But  kdx  is  the  differential  of  y;  A'dx  is  the  differential  of 
8 ;  and  A"dx  is  the  differential  of  v.     Hence 
du=dy+dz  —  do. 
Ex.  1.  What  is  the  differential  of  Gxi-5xi-2x? 

Ans.  (24^8-l5^-2)^. 
Ex.  2.  What  is  the  differential  of  ax*— ex? 

Ans. 
Ex.  3.  What  is  the  differential  of  3ax3-bx*  ? 

Ans. 
Ex.  4.  What  is  the  differential  of  a6+3aV+3aV+^6? 

Ans. 
Ex.  5.  What  is  the  differential  of  5.r3-3x2+6x+2? 

Ans. 
Ex.  6.  What  is  the  differential  of  7.;r5+6r,-5«:r  +  3.r-6  ? 

J.  715'. 

Proposition  IX. — Theorem. 
(180.)   77*e  differential  of  the  product  of  two  functions  de- 
pendent on  the  same  variable,  is  equal  to  the  sum  of  the  products 
obtained  by  multiplying  each  by  the  differential  of  the  other. 

Let  us  designate  two  functions  by  u  and  v,  and  suppose 
them  to  depend  on  the  same  variable  x ;  then,  when  x  is  in- 
creased so  as  to  become  x+h,  the  new  functions  may  be  writ- 
ten, Art.  178, 

u'=u+Ah  +  B/r, 
v'=v+A'h  +  B'h\ 
If  we  multiply  together  the  corresponding  members  of  these 
equations,  we  shall  have 

u'v'  =  uv  +  Avh  +Bvh\ 

+  A'uh  +  AA'h2+,  etc., 

+  B'uh*  +,  etc., 

where,  it  will  be  observed,  the  terms  omitted  contain  powers  of 

the  increment  higher  than  h\ 

Transposing,  and  dividing  by  //,  w'e  have 

u'v'—uv 

t =Av+A'u+  other  terms  involving  h. 

When  we  make  h  equal  to  zero,  Art.  174,  the  terms  involving 
h  disappear,  and  we  have 


128  Differential   Calculus. 

d(uv) 

-±-!-  =  Av+Alu; 
ax 

or,  multiplying  by  dx, 

d(u  v)  =  vAdx + uA'dx. 

But  Adx  is  equal  to  du, 

and  A'dx  is  equal  to  dv. 

Hence  d(uv)  =  vdu+udv,  (1) 

which  was  the  proposition  to  be  demonstrated. 

(181.)  Cor.  If  we  divide  both  members  of  equation  (1)  by 
uv,  we  shall  have 

d(uv)     du    dv 

UV  U         V 

that  is,  the  differential  of  the  product  of  two  functions,  divided 
by  their  product,  is  equal  to  the  sum  of  the  quotients  obtained 
by  dividing  the  differential  of  each  function  by  the  function 
itself. 

Ex.  1.  What  is  the  differential  ofxy'l 

Ans.  y*dx  +  2xydy. 
Ex.  2.  What  is  the  differential  ofx'if! 

Ans. 
Ex.  3.  What  is  the  differential  of  ax2y*  ? 

Ans. 
Ex.  4.  What  is  the  differential  of  ax*(x*+2b)  ? 

Ajis. 
Ex.  5.  What  is  the  differential  of  (.r3+«)  (2x+b)  ? 

Ans. 
Ex.  G.  What  is  the  differential  of  (x3+a)  (3x'+b)  ? 

Ans. 

Proposition  X. — Theorem. 

(1S2.)  The  differential  of  the  product  of  any  number  of  func- 
tions of  the  same  variable,  is  equal  to  the  sum  of  the  products 
obtained  by  multiplying  the  differential  of  each  function  by  the 
product  of  the  others. 

Let  us  designate  three  functions  by  u,  v,  and  z,  and  suppose 
them  to  depend  on  tre  same  variable  x.  Substitute  y  for  vz, 
and  we  shall  have 


Differentiation    of   Algebraic    Functions.  129 

uvz=uy, 
ind  d(uvz)  =  d{uy). 

But,  by  the  preceding  Proposition, 

d(uy)—ydu+udy ;  (1) 

and  since  y=vz,  we  have,  by  the  same  Proposition, 
dy=zdv+vdz. 
Substituting. these  values  of  y  and  dy  in  equation  (1),  it  be- 
comes 

d(iivz)  =  vzdu+uzdv+uvdz.  (2) 

The  same  method  is  applicable  to  the  product  of  four  or 
more  functions. 

(183.)   Cor.  If  we  divide  both  members  of  equation  (2).  by 
uvz,  we  shall  have 

d'(uvz)_du    dv    dz 
uvz        uvz 
which  is  an  extension  of  Art.  181. 

Ex.  1.  What  is  the  differential  of  xy2z? 

Ans.  y2zdx+2xyzdy+xy'dz. 

Ex.  2.  What  is  the  differential  ofan/V? 

Ans.  ■ 

Ex.  3.  What  is  the  differential  of  axy'z3  ? 

Ans. 
Ex.  4.  What  is  the  differential  of  x{x^a)  (x+2b)  ? 

Ans. 
Ex.  5.  What  is  the  differential  of  ax\x*+a)  (x+3b)  ? 

Ans. 

Proposition  XI. — Theorem. 
(184.)    The  differential  of  a  fraction  is  equal  to  the  denom- 
inator into  the  differential  of  the* numerator,  minus  the  numer- 
ator into  the  differential  of  the  denominator,  divided  by  the  square 
of  the  denominator. 

Let  us  designate  the  fraction  by  -,  ard  suppose 

'--=y,  (1)      • 

v 

then  u=vy- 

I 


130  Differential   Calculus. 

Therefore,  by  Prop.  IX., 

du=ydv  +  vdy ; 
whence  vdy=du—ydv.  (2) 

Substituting  in  the  second  member  of  equation  (2)  the  value 
:>f  y  from  equation  (1),  we  have 

udv 
vdy=du . 

Dividing  by  v,  we  obtain 

vdu—udv 


dy 
that  is,  d(-) 


u\     vdu—udv 


v 
which  was  the  proposition  to  be  demonstrated. 

(185.)   Cor.  If  the  numerator  u  is  constant,  its  differentia 
will  be  zero,  Art.  177,  and  we  shall  have 


c\      —cdv 
v)~     d2 


x 
Ex.  1.  What  is  the  differential  of  —  ? 


y 

■2xi/sdx—3x~y*dy        2xydx  —  3x*dw 
Ans.  — — ; — Z—Z-,  or  — - ; -, 


Ex.  2.  What  is  the  differential  of  —  ? 

xa 

Ans. 

Ex.  3.  What  is  the  differential  of  — 3? 

ax 

Ans. 

Ex.  4.  What  is  the  differential  of  -^—  ? 

l—x 

Ans. 

1+x* 
Ex.  5.  What  is  the  differential  of — ^? 

l—x 

Ans. 

a*+x' 

Ex.  6.  What  is  the  differential  of  ti s? 

0  —x 

Ans. 


\ 


Differentiation   of   Algebraic   Functions.INI 

Proposition  XII. — Theorem. 

(186.)  To  obtain  the  differential  of  a  variable  affected  with 
any  exponent  whatever,  we  must  diminish  the  exponent  of  the 
power  by  unity,  and  then  multiply  by  the  primitive  exponent 
and  by  the  differential  of  the  variable. 

This  is  the  same  as  Prop.  IV.,  and  the  demonstration  there 
given,  being  founded  on  the  binomial  theorem,  may  be  con- 
sidered sufficiently  general,  since  the  binomial  theorem  is  true, 
whether  the  exponent  of  the  power  be  positive  or  negative, 
integral  or  fractional.  This  theorem  may,  however,  be  de- 
duced directly  from  Prop.  X. 

Let  it  be  required  to  find  the  differential  of  re",  where  the  ex- 
ponent n  may  be  either  positive  or  negative,  integral  or  frac 
tional. 

Case  first.  When  n  is  a  positive  and  whole  number. 

xn  may  be  considered  as  the  product  of  n  factors  each  equai 
to  x.     Hence,  by  Prop.  X.,  Cor., 

d(xn)     dixxxx  .  .  .  .)     dx    dx    dx     dx 

X  xxxx  ....         X        X        X        X 

and  since  there  are  n  equal  factors  in  the  first  member  of  the 
equation,  there  will  be  n  equal  terms  in  the  second ;  hence 

d{xn)     ndx 

or  d(xn)  =  nxn~xdx. 

Case  second.  When  n  is  a  positive  fraction. 

T 

Represent  the  fraction  by  -,  and  let 

r 

u—x3. 

Raising  both  members  to  the  power  s,  we  shall  have 

u°=xT, 

and,  since  r  and  s  are  supposed  to  represent  entire  numbers, 

we  shall  hctve,  by  the  first  Case, 

s  u*~ldu = 7*xT~i  dx  ; 

rxr~1  rxr~l 

whence  we  find       du= — —rdx—-^ — dx. 

SU  -(.-1) 

sx" 
which  may  be  reduced  to 


132  Differential  Calculus. 

du  —  -xs    dx, 
s 

f 

wiiich  is  of  the  same  form  as  nx*~ldx,  substituting  -  for  n. 

°  s 

Case  third.  When  n  is  negative,  either  integral  or  fractional 
Suppose  u=x~", 

which  may  be  written  u= — . 

J  xa 

Differentiating  by  Prop.  XL,  Cor.,  we  have 

x2a      ' 

and  differentiating  the  numerator  by  Case  first,  or  by  Case 

second,  if  n  represents  a  fraction,  we  have 

—  nxn~1dx 
du= -^ ; 

or,  subtracting  the  exponent  2n  from  n—  1,  we  have 

du=—nx~D~1dx, 
which  is  of  the  same  form  as  nxn~ldx,  by  substituting  —  n  for  -f?i, 
Proposition  XII.  may,  therefore,  be  considered  general,  what 
ever  be  the  exponent  of  a\ 

Ex.  1.  What  is  the  differential  of  ax**1? 

Ans.  a(n+l)xndx. 

a  - 
Ex.  2.  What  is  the  differential  of  -xn+c? 

b 

Ans. 

Ex.  3.  What  is  the  differential  of  aVx^l 

Ans. 

Ex.  4.  What  is  theMifferential  of  far*? 

Ans. 
Ex.  5.  What  is  the  differential  of  cx~z1 

Ans. 

Ex.  6.  What  is  the  differential  of  x'2y2za  ? 

Ans. 
Ex.  7.  If  the  area  of  a  square  increase  uniformly  at  the  rate 
of  r\  of  a  square  inch  per  second,  at  what  rate  is  the  side  in- 
creasing when  the  area  is  100  square  inches? 

Ans. 


Differentiation    of    Algebraic   Functions.  133 

Ex  8.  If  the  solidity  of  a  cube  increase  uniformly  at  the 
rate  of  a  cubic  inch  per  second,  at  what  rate  is  the  edge  in- 
creasing  when  the  solid  becomes  a  cubic  foot  ? 

Ans. 

Proposition  XIII. — Theorem. 
(187  )   The  differential  of  the  square  root  of  a  variable  quan- 
tity, is  equal  to  the  differential  of  that  quantity  divided  by  twice 
the  radical. 

Let  it  be  required  to  find  the  differential  of 

y/x,  or  x2. 
According  to  the  preceding  Proposition, 

i  i-i 

d(x2)  =  \x2    dx, 
i 
=  \x  2dx, 

dx 
which  may  be  written  ~%Jx 

Ex.  1.  What  is  the  differential  of  Vax*1 

Sax^dx         .  i  f  . 
Ans.  — =,  or  %a~x-dx. 

2Vax* 

Ex.  2.  What  is  the  differential  of  Vabx71 

Ans. 

Ex.  3.  What  is  the  differential  of  yTax'l 

Ans. 

—    x 
Ex.  4.  What  is  the  differential  of  aVx— -  I 

Ans. 

Ex.  5.  What  is  the  differential  of  V~ax+  -/cV? 

Ans. 

Proposition  XIV. — Theorems 
(188  )   To  obtain  the  differential  of  a  polynomial  raised  to  any 
power,  we  must  diminish  the  exponent  of  the  power  by  unity, 
and  then  multiply  by  the  primitive  exponent  and  by  the  differ- 
ential of  the  polynomial. 

Let  it  be  required  to  differentiate  the  function 
u=(ax+xy. 


134  Differential   Calculus, 

Substitute  y  for  ax+x*,  and  we  have 

u=y\ 
Whence,  by  Prop.  XII.,      du=nyn~1dy. 
Restoring  the  value  of  y,  we  have 

du — n  {ax  -\-x"Y~ld{ax +£a) , 
which  is  conformable  to  the  Proposition. 

The  differentiation  of  ax+x*  is  here  only  indicated.     Ii  «ve 
nctually  perform  it,  we  shall  have 

du — n  {ax + x2) n_I  (a + 2x)  dx. 


Ex.  1.  What  is  the  differential  of  Va+bx2? 

bxdx 
Ans. 


'a+bx' 

Ex.  2.  What  is  the  differential  of  {ax*+x*)3  ? 

Ans. 


Ex.  3.  What  is  the  differential  of  Vax+bx^+cx3! 

Ans. 
Ex.  4.  What  is  the  differential  of  {ax-xyi 

Ans. 

Ex.  5.  What  is  the  differential  of  {a+bxrf  ? 

Ans. 
i 
Ex.  6.  What  is  the  differential  of  {a+x*)~l 

Aiis. 
Ex.  7.  If  x  increase  uniformly  at  the  rate  of  T^  of  an  inch 
per  second,  at  what  rate  is  the  expression  (l+#)3  increasing 
when  x  equals  9  inches  ?  Ans. 

(189.)  By  the  application  of  the  preceding  principles,  com- 
plicated algebraic  functions  may  be  differentiated. 

a~\-x 

Ex.  8.  What  is  the  differential  of  the  function  u= 5  ? 

a+x 

According  to  Prop.  XI., 

(a + x*)  dx — 2x  {a + x)  dx 

{a+xy 

which  may  be  reduced  to 

{a— 2ax— x*)dx 


du- 


{a+xj 


Ex.  9.  Differentiate  the  function  u—  Vx^+a^/x. 

Ans. 


Differentiation   of  Algebraic  Functions.  133 

(b+xY 


Ex.  10.  Differentiate  the  function  u  =  - 


Ex.  11.  Differentiate  the  function  u= 


x 
Ans. 

x1 

(a+xj 

Ans. 


Ex.  12.  Differentiate  the  function  u=- 


(a+x2)a' 
Ans. 
Ex.  13.  If  the  side  of  an  equilateral  triangle  increase  uni- 
formly at  the  rate  of  half  an  inch  per  second,  at  what  rate  is 
its  perpendicular  increasing  when  its  side  is  equal  to  8  inches? 

Ans.  — —  inch  per  second. 
4 

Ex.  14.  If  the  side  of  an  equilateral  triangle  increase  uni- 
formly at  the  rate  of  half  an  inch  per  second,  at  what  rate  is 
the  area  increasing  when  the  side  becomes  8  inches  ? 

Ans.  2v/3  inches  per  second. 

Ex.  15.  If  a  circular  plate  of  metal  expand  by  heat  so  that 
its  diameter  increases  uniformly  at  the  rate  of  T\-$  of  an  inch 
per  second,  at  what  rate  is  its  surface  increasing  when  the  di- 
ameter is  exactly  two  inches  ? 

Ans.  — —  inch  per  second 

Ex.  16.  If  a  circular  plate  expand  so  that  its  area  increases 
uniformly  at  the  rate  of  T\  of  a  square  inch  per  second,  at  what 
rate  is  its  diameter  increasing  when  the  area  of  the  circle  is 
exactly  a  square  inch? 

Ans.  — inch  per  second. 

50-v/TT  r 

Ex.  17.  If  the  diameter  of  a  spherical  soap  bubble  increases 
uniformly  at  the  rate  of  T\  of  an  inch  per  second,  at  what  rate 
is  its  capacity  increasing  at  the  moment  the  diameter  becomes 
two  inches  1 

Ans.  -  inch  per  second. 
o 

Ex.  18.  If  the  capacity  of  a  spherical  soap  bubble  inci  eases 
uniformly  at  the  rate  of  two  cubic  inches  per  second,  at  what 


136  Differential   Calculus. 

rate  is  the  diameter  increasing  at  the  moment  it  becomes  two 
inches? 

Ans.  -  inch  per  second. 

IT 

Ex.  19.  A  boy  standing  on  the  top  of  a  tower  whose  height 
is  60  feet,  observed  another  boy  running  toward  the  foot  of  the 
tower  at  the  rate  of  five  miles  an  hour  on  the  horizontal  plane  ; 
at  what  rate  is  he  approaching  the  first  when  he  is  80  feet  from 
the  foot  of  the  tower  ? 

Ans.  4  miles  an  hour. 

Ex.  20.  If  the  diameter  of  a  circular  plate  expand  uniformly 
at  the  rate  of  T\  of  an  inch  per  second,  what  is  the  diameter 
of  the  circle  when  its  area  is  expanding  at  the  rate  of  a  square 
inch  per  second  1 

Ans.  —  inches. 

7T 

Ex.  21.  If  the  diameter  of  a  sphere  increase  uniformly  at  the 
rate  of  r\  of  ah  inch  per  second,  what  is  its  diameter  when 
the  capacity  is  increasing  at  the  rate  of  five  cubic  inches  per 
second  ? 

Ans. inches. 

Ex.  22.  If  the  diameter  of  the  base  of  a  cone  increase  uni- 
formly at  the  rate  of  Ty  inch  per  second,  at  what  rate  is  its 
solidity  increasing  when  the  diameter  of  the  base  becomes  10 
inches,  the  height  being  constantly  one  foot? 

Ans.  2tt  inches  per  second. 


SECTION   II. 

OF  SUCCESSIVE  DIFFERENTIALS  —  MACLAURTN'S  THEOREM  — 
TAYLOR'S  THEOREM  — FUNCTIONS  OF  SEVERAL  INDEPEND- 
ENT  VARIABLES. 

(190.)  Since  the  differentials  of  all  expressions  which  con- 
tain x  raised  to  any  power,  also  contain  x  raised  to  the  next 
inferior  power,  Art.  186,  we  may  consider  the  differential  co- 
efficient of  a  function  as  a  new  function,  and  determine  its  dif- 
ferential accordingly.  We  thus  obtain  the  second  differential 
coefficient. 

For  example,  if  tc=ax3, 

du 

-rr-=2ax . 
dx 

Now  since  3a«2  contains  x,  we  may  differentiate  it  as  a  new 
function,  and  we  obtain 

d[~~r)  =Gaxdx. 

But,  since  dx  is  supposed  to  be  a  constant, 

ydu\_d(du)     d2u 
\dxJ        dx       dx  ' 

the  symbol  d*u  (which  is  read  second  differential  of  u)  being 

used  to  indicate  that  the  function  u  has  been  differentiated 

twice,  or  that  we  have  taken  the  differential  of  the  differential 

oftt.     Hence 

*  d*u 

-7—  =6axdx : 
dx 

or,  dividing  each  side  by  dx, 

d2u 

dV=Gax> 

wnere  dx"2  represents  the  square  of  the  differential  of  x,  and  noi 
the  differential  of  z\ 

The  expression  Gax  being  the  differential  coefficient  of  the 


-6a, 


138  Differential   Jalculus. 

first  differential  coefficient,  is  called  the  second  differential  co- 
efficient. 

Again,  since  Gax  contains  x,  we  may  differentiate  it  as  a 
new  function,  and  we  obtain 

d3u 

— =6adx  ; 

or,  dividing  each  side  by  dx, 

dSi 

dx3 

which  is  the  differential  coefficient  of  the  second  differential 

coefficient,  and  is  called  the  third  differential  coefficient. 

d3u  . 
The  third  differential  coefficient  -=-;  is  read  third  differential 

dx3 

of  m,  divided  by  the  cube  of  the  differential  of  x. 

As  the  expression  6a  does  not  contain  x,  the  differentiation 
can  be  carried  no  further,  and  we  find  the  function  u=ax3  has 
three  differential  coefficients.  Other  functions  may  have  a 
greater  number  of  differential' coefficients. 

The  learner  must  not  confound  d2u  with  oV,  the  former  de- 
noting the  differential  of  the  differential  of  u,  and  the  latter  the 
square  of  the  differential  of  u. 

Ex.  1.  Determine  the  successive  differentials  of  ax4. 

Ex.  2.  Determine  the  successive  differentials  of  (a+.r2)3. 

MACLAURIN'S  THEOREM. 

(191.)  Maclaurin's  theorem  explains  the  method  of  develop- 
ing into  a  series  any  function  of  a  single  variable. 

Proposition  I. — Theorem  of  Maclaurin. 

If  u  represent  a  function  of  x  which  it  is  possible Jp  develop  in 
a  series  of  positive  ascending  powers  of  that  variable,  then  will 
that  developinent  be 

,  v      (du\        (d?u\x*     (d3u\  x3 
^^+U)^(avJ2  +  (av)^+'etC" 
where  the  brackets  indicate  the  values  which  the  inclosed  func- 
tions assume  when  x  equals  zero. 

Let  u  represent  any  function  of  x,  as,  for  example,  (a+x)n, 
and  let  us  suppose  that  this  function,  when  expanded,  will  con- 


Maclauein's   Theorem.  139 

tain  the  ascending  powers  of*,  and  coefficients  not  containing 
x,  which  are  to  be  determined.  Let  these  coefficients  be  rep- 
resented by  A,  B,  C,  etc.,  then  we  shall  have 

u=A+Bx+Cx-+T>x'+'&xi+,  etc.        (1) 
If  we  differentiate  this  equation,  and  divide  both  sides  by  dx, 
we  obtain 

— =B+2Cx+3T)x2+4Vxi+,  etc. 
dx 
If  we  continue  to  differentiate,  and  divide  by  dx,  it  is  ob 
vious  that  the  coefficients  A,  B,  C,  etc.,  will  disappear  in  suc- 
cession, and  the  result  will  be  as  follows : 

^=2C+2.3D.r+    3.4Ez2+,  etc., 
dx2 

—=         2.3D  +2.3.4E.r  +,  etc., 
dx3 

etc.,  etc. 

Represent  by  (u)  what  u  becomes  when  x=0. 

Represent  by  (^)  what  ^  becomes  when  x=0. 

Represent  by  (^)  what  -^  becomes  when  x=0, 

and  so  on ;  the  preceding  equations  furnish  us 

(u)     =A, 

(§)  =B' 

(S)=-° • 

whence  we  see 

Substituting  these  values  in  equation  (1),  it  becomes 

which  is  Maclaurin's  theorem. 

(192.)  Ex.  1.  Expand  (a+x)n  into  a  series. 

When  .r=0,  this  function  reduces  to  an. 
Hence  (u)  =  a\ 


140  Differential   Calculus. 

By  differentiation,  we  obtain 

du       ,        .     , 
s=«(«+x)-f 

which  becomes,  when  £=0, 

Hence  (JLj—na»-im 

Also,  ^i=n(n-l)  («+*)*-, 

which  becomes,  when  x=0, 

w(t7.—  l)an_2. 
ai  dsu 

'  d?=n{-n~ l)  (n_2)  (a+x)n~3> 

which  becomes,  when  x=0, 

n(n-l)(n-2)an-\ 
Substituting  these  values  in  Maclaurin's  theorem,  we  have 

(a+xy=an+nan-lx+n{n~    a"-y+,  etc., 
which  is  the  same  as  found  by  the  Binomial  theorem. 

Ex.  2.  Develop  into  a  series  the  function  u= . 

a+x 

By  differentiation,  we  find 

d  u  _  1 

dx  (a+xy' 

(£u_  2 

dx*~  (a+x)3' 

d3u_  2.3 

dx3^~{a+xy' 

Making  x=0,  in  the  values  of  u,  of  -z-,  of  -r-r,  of  -r-„  etc.,  we 

ax        ax         ax 

find 

,  .      1     fdu\  1      fd2u\      2     fd3u\         2.3 

(M)=v  UJ=-^'  U?J=^  WvH— »etc- 

Substituting  these  values  in  Maclaurin's  theorem,  we  obtain 
1        1     x     x2    x3 

:~~^+7i-^T+'  etC- 


a-\x    a     a*     a3     a* 


Taylor's   Theorem.  141 

Ex.  3.  Develop  into  a  series  the  function . 

r  1—x 

Ans.  l+x+x*+xs+x*+,  etc. 


Ex.  4.  Develop  into  a  series  the  function  Va+x. 

i     1   -i         1-3  1.3    -5 

.Ans.  a2+-a  2x— — a  2xM a  2x3— ,  etc. 

2  2.4  2.4.6  ' 

Ex.  5.  Expand  into  a  series  (a  —  x)~2. 

Ans. 

Ex.  6.  Expand  into  a  series  (a+x)~3. 

Ans.  a-3— 3a_4a;+6a-V— 10a-V+15a-V-,  etc. 

(193.)  When  in  the  application  of  Maclaurin's  theorem  the 

variable  x  is  made  equal  to  0,  the  function  u,  or  some  of  its 

differential  coefficients,  may  become  infinite.     Such  functions 

can  not  be  developed  by  Maclaurin's  theorem. 

Thus,  if  we  have 

1 

w=log.  x,  u=coi.  x,  or  u—-, 

°  X 

when  we  make  x=0,  u  becomes  equal  to  infinity. 

Also,  if  we  have  u=ax2, 

the  first  differential  coefficient  is 

du      a 

dx=     V 

ax    2x2 

vvhich  becomes  infinite  when  x  is  made  equal  to  zero. 

Hence  neither  of  these  functions  can  be  developed  by  Miio 
laurin's  theorem. 

TAYLOR'S  THEOREM. 

(194.)  Taylor's  theorem  explains  the  method  of  developing 
into  a  series  a  function  of  the  sum  or  difference  of  two  variables. 

The  following  principle  is  assumed  in  the  demonstration  of 
Tavlor's  theorem. 

Proposition  II. — Theorem. 

If  we  have  a  function  of  the  sum  or  difference  of  two  varia- 
bles, x  and  y,  the  differential  coefficient  will  be  the  same,  if  we 
uppose  x  to  vary  and  y  to  remain  constant,  as  when  we  suppose 
y  to  vary  and  x  to  remain  constant. 

Thus,  let  u  =  (x+y)\ 


f 42  Differential   Calculus. 

If  we  suppose  x  to  vary  and  y  to  remain  constant,  we  have 

du       ,        .    . 

-=n(x+y)-; 

and  if  we  suppose  y  to  vary  and  x  to  remain  constant,  we  have 
du  :     . 

the  same  as  under  the  first  supposition. 

Proposition  III. — Taylor's  Theorem. 

(195.)  .Any  function  of  the  sum  of  two  variables  may  be  de- 
veloped into  a  series  of  the  following  form : 

„,        .  du       d2u  v2      d?u    v3 

^+y)=*+5y+5?  t2+te-UTS+' etc- 

where  u  represents  the  value  of  the  function  when  y=0. 

Let  u'  be  a  function  of  x+y,  which  we  will  suppose  to  be 
developed  into  a  series,  and  arranged  according  to  the  powers 
of  y,  so  t-hat  we  have 

it'=F(x+y)=A+By+Cy*+T)y3+,  etc.,     (1) 

where  A,  B,  C,  D,  etc.,  are  independent  of  y,  but  dependent 
upon  x  and  upon  the  constants  which  enter  the  primitive  func- 
tion. It  is  now  required  to  find  such  values  for  A,  B.  C,  D, 
etc.,  as  shall"  render  the  development  true  for  all  possible 
values  which  may  be  attributed  to  x  and  y. 

If  we  differentiate  under  the  supposition  that  x  varies  and  y 
remains  constant,  we  shall  have 

dtuJ__dA    dB      dC  a    dB  3 
dx     dx     dxJ    dx         dx 

If  we  differentiate  under  the  supposition  that  y  varies  and  x 
remains  constant,  we  shall  have 

du'    _  '     _         _   . 
-3-=B+2Cy+3D2/2+,  etc. 

But,  by  the  preceding  Proposition, 
du'  _du' 
dx     dy' 
Hence  we  must  have 
dA    dB      dC 
~dx~+~dx~y+dx~!/''i~>  etC"  =B+2Cy+3D</'+,  etc., 


Taylor's    Theorem. 

and  since  the  coefficients  of  the  series  are  independent  of  y. 
and  the  equality  exists  whatever  be  the  value  of  y,  it  follow? 
that  the  terms  involving  the  same  powers  oft/  in  the  two  mem 
bers  are  respectively  equal  (Algebra,  Art.  300).     Therefore, 

^=B,  (2) 

ax 

^=2C,  (3) 

ax 

£=3D,  etc.  (4) 

ax 

If  in  equation  (1)  we  make  y=0,  function  of  x+y  will  reduce 

to  function  of  x,  which  we  will  denote  by  u.     Therefore 

A=u. 

Substituting  this  value  of  A  in  equation  (2),  we  have 

B=— 

dx 

Substituting  this  value  of  B  in  equation  (3),  we  have 

i  (Fu  ^     d?u 

2C=^;  whence  C=^. 

Substituting  this  value  of  C  in  equation  (4),  we  have 

3D  =  2^;  whenCeD=2"3^- 

Substituting  these  values  of  A,  B,  C,  D  in  equation  (1),  we 

have  Taylor's  formula, 

du      d2u  y1   ,  dht    y* 
ul=F{x+y)=u+Txy+-  -+^  — +,  etc., 

where  the  first  term  of  the  series,  u,  represents  what  the  func- 
tion to  be  developed  becomes  when  the  variable,  according  to 
•  the  ascending  powers  of  which  the  series  is  arranged,  is  made 
equal  to  zero. 

In  a  similar  manner,  we  find  the  development  of  F(x-y) 

to  be 

du      d2u  \f     <Fu    y3 
«'=Ffe-y)=«-^+^ii-^i^3+  etc- 

(196.)  Ex.  1.  Required  the  development  of  the  function 

a'=(a?+y)°- 
Making  y=0,  we  obtain  u=x\  and  thence,  by  differentiation. 


144  Differential.  Calculus. 

du  ,    d*u 

Tx=nX  ".&=*(*-!)*  "-*> 

— =7i(n-l)(n-2)xa-\  etc. 

These  values  being  substituted  in  the  formula,  give 
s  ,      n(n—  1)  n(n—  1)  (n  —  2) 

etc. 

the  same  as  found  by  the  Binomial  theorem. 

Ex.  2.  Required  the  development  of  the  function 
%'=  Vx+y. 
i        i      1    -1  1     _Ji  1.3     _i 

Ans.  u,=(z+y)*=xz+-x  2y~^  y+^-^-c  y-,etc 

Ex.  3.  Required  the  development  of  the  function 
u'=Vx-\-y. 

i       a     1  _a        2-a         2.5    _i 
Ans.  u'=(x+y)3=x3+-x  zy~^  Y+gg-g*  Y".  etc. 

(197.)  Although  the  genera*!  development  of  every  function 
of  x+y  is  correctly  given  by  Taylor's  theorem,  particular 
values  may  sometimes  be  assigned  to  the  variables  which  shall 
render  this  form  of  development  impossible;  and  this  impos- 
sibility will  be  indicated  by  some  of  the  coefficients  in  Taylor's 

theorem  becoming  infinite.     Thus,  if  we  have 

i 
u'=a-\-(b— x+  y)2, 

u  =a-jr(b—x)3. 

Therefore,  — = 

dx        2{b-xf 
<Tu  1 

ax  4(b-x)2 

etc.,       etc., 

in  which  all  the  differential  coefficients  will  become  equal  to 
infinity,  when  we  make  x—b. 
So,  also,  if  we  have 

u'=a+(b— x+y)a, 
in  which  n  is  a  whole  number,  all  the  differential  coefficients 
will  become  infinite  when  we  make  x=b. 

The  supposition  that  x-=b  reduces  the  above  equation  to 


Differentiation   of   Functions.  145 

i 
u'— u+yn> 
and  u  =a, 

where  u'  and  u  are  expressed  under  different  forms  ;  and,  in 
general,  when  the  proposed  function  changes  its  form  by  at- 
tributing particular  values  to  the  variables,  the  development 
can  not  be  made  by  Taylor's  theorem. 

DIFFERENTIATION  OF  FUNCTIONS  OF  TWO  OR  MORE  INDEPEND- 
ENT VARIABLES. 

(198.)  Let  u  be  a  function  of  two  independent  variables  x 
and  y ;  then,  since  in  consequence  of  this  independence,  how- 
ever either  be  supposed  to  vary,  the  other  will  remain  un- 
changed, the  function  ought  to  furnish  two  differential  coeffi- 
cients ;  the  one  arising  from  ascribing  a  variation  to  x,  and  the 
other  from  ascribing  a  variation  toy;  y  entering  the  first  co- 
efficient as  if  it  were  a  constant,  and  x  entering  the  second  as 
if  it  were  a  constant. 

If  we  suppose  y  to  remain  constant  and  x  to  vary,  the  dif- 
ferential coefficient  will  be 

du 
dx ' 
and  if  we  suppose  x  to  remain  constant  and  y  to  vary,  the  de- 
ferential coefficient  will  be 

du 

dy 

(199.)  The  differential  coefficients  which  are  obtained  under 

these  suppositions  are   called  partial  differential  coefficients. 

The  first  is  the  partial  differential  coefficient  with  respect  to  x, 

and  the  second  with  respect  to  y. 

If  wc  multiply  the  several  partial  differential  coefficients  by 
dx  and  dy,  we  obtain 

du        du 

dx**'  iydy> 

which  are  called  partial  differentials ;  the  first  is  a  partial  dif- 
ferential with  respect  to  x,  and  the  second  a  partial  differential 
with  respect  to  y. 

The  differential  which  is  obtained  under  the  supposition  that 
both  the  variables  have  changed  their  values,  is  called  the  total 
differential  of  the  function  ;  that  is, 
*  K 


146  Differential   Calculus. 

du         dh, 
du = —dx  H — —d  v. 

dx         dy  J 

(200.)  If  we  have  a  function  of  three  variables,  x,  y,  and  z, 
we  should  necessarily  have  as  many  independent  differentials, 
of  which  the  aggregate  would  be  the  total  differential  of  the 
function  ;  that  is, 

.       die         du        du . 
du  =  -j-dx  ■\--rdii+ -r-dz. 
dx         dy  J     dz 

Hence,  whether  the  variables  are  dependent  or  independent, 
we  conclude  that  the  total  differential  of  a  function  of  any  num- 
ber of  variables  is  the  sum  of  the  several  partial  differentials, 
arising  from  differentiating  the  function  relatively  to  each  varia- 
ble in  succession,  as  if  all  the  others  were  constants. 

Ex.  1.  If  one  side  of  a  rectangle  increase  at  the  rate  of  1 
inch  per  second,  and  the  other  at  the  rate  of  2  inches,  at  what 
rate  is  the  area  increasing  when  the  first  side  becomes  8  inches, 
and  the  last  12  inches? 

vlns.  28  inches  per  second. 

Ex.  2.  If  one  side  of  a  rectangle  increase  at  the  rate  of  2 
inches  per  second,  and  the  other  diminish  at  the  rate  of  3  inches 
per  second,  at  what  rate  is  the  area  increasing  or  diminishing, 
when  the  first  side  becomes  10  inches,  and  the  second  8? 

Ans. 

Ex.  3.  If  the  major  axis  of  an  ellipse  increase  uniformly  at 
the  rate  of  2  inches  per  second,  and  the  minor  axis  at  the  rate 
of  3  inches,  at  what  rate  is  the  area  increasing  when  the  major 
axis  becomes  20  inches,  the  minor  axis  at  the  same  instant  be 
ing  12  inches  ? 

Ans.  21tt  inches. 

Ex.  4.  If  the  altitude  of  a  cone  diminishes  at  the  rate  of  3 
inches  per  second,  and  the  diameter  of  the  base  increases  at 
the  rate  of  1  inch  per  second,  at  what  rate  does  the  solidity 
vary  when  the  altitude  becomes  18  inches,  the  diameter  of  the 
base  at  the  same  instant  being  10  inches? 

Ans. 


R' 
Then 


SECTION    III. 

SIGNIFICATION   OF   THE   FIRST   DIFFERENTIAL   COEFFICIENT- 
MAXIMA  AND  MINIMA  OF  FUNCTIONS. 

Proposition  I. — Theorem. 
(201.)   The  tangent  of  the  angle  which  a  tangent  line  at  any 
point  of  a  curve  makes  with  the  axis  of  abscissas,  is  equal  to  the 
first  differential  coefficient  of  the  ordinate  of  the  curve. 

Let  CPP'  be  a  curve,  and  P  any 
point  of  it  whose  co-ordinates  are 
x  and  y.  Increase  the  abscissa 
CR  or  x  by  the  arbitrary  incre- 
ment RR',  which  we  will  repre- 
sent  by  h ;  denote  the  correspond- 
ing ordinate  PR'  by  y',  and  draw  the  secant  line  SPP. 
PD=P,R'-PR=3/'-y- 

But  from  the  triangle  PDP'  we  have 

PD  :PD::  1  :  tang.  S=p^; 

and,  substituting  for  PD  and  PD  their  values,  we  have 

^-^=tang.  S, 
ft  3 

which  expresses  the  ratio  of  the  increment  of  y  to  that  of  x. 
In  order  to  find  the  differential  coefficient  of  y  with  respect  to 
t,  we  must  find  the  limit  of  this  ratio  by  making  the  incremen 
equal  to  zero,  Art.  174. 

Now  if  ft  be  diminished,  the  point  P'  approaches  P,  the  secant 
SP  approaches  the  tangent  TP ;  and,  finally,  when  ft=0,  the 
point  P'  coincides  with  P,  and  the  secant  with  the  tangent. 
In  this  case  we  have 

-p  =  tang.  1. 
ax 

(202.)  If  it  is  required  to  find  the  point  of  a  given  curve  at 


14S 


Differential   Calculus. 


which  the  tangent  line  makes  a  given  angle  with  the  axis  of  X, 
we  must  put  the  first  differential  coefficient  equal  to  the  tangent 
of  the  given  angle.  If  we  represent  this  tangent  by  a,  we 
must  have  * 

dy 

ax 
and  this,  combined  with  the  equation  of  the  curve,  will  give  the 
values  of  a;  and  y  for  the  required  point.     ■ 

Ex.  It  is  required  to  find  the  point  on  a  parabola,  at  which 

the  tangent  line  makes  an  angle  of  45°  with  the  axis. 

The  equation  of  the  parabola,  Art.  50,  is 

y2=2px. 

Differentiating,  we  obtain 

2ydy—2pdx, 

dy     p 
or  -r=—. 

dx     y 

But  since  tang.  45°  equals  radius  or  unity,  we  have 

P_ 

y~ 

Whence,  from  the  equation  y"~=2px,  we  find 

-,-P 
X~2' 

Hence  the  required  tangent  passes  through  the  extremity  ot 
the  ordinate  drawn  from  the  focus. 


1,  or  p=y. 


OF  THE  MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  A  SINGLE 
VARIABLE. 

(203.)  If  a  variable  quantity  gradually  increase,  and,  after  it 
has  reached  a  certain  magnitude,  gradually  decrease,  at  the 
end  of  its  increase  it  is  called  a  maximum. 

Thus,  if  a  line  P'R',  moving  from  A 
along  AB  so  as  to  be  always  at  right 
angles  to  AB,  gradually  increases  until 
it  comes  into  the  position  PR,  and  after 
that  gradually  decreases,  the  line  is 
said  to  be  a  maximum,  or  at  its  greatest   "  R'  R  R" 

value,  when  it  comes  into  the  position  PR. 

(204.)  If  a  variable  quantity  gradually  decrease  and,  after 


Maxima    and   Minima   of   Functions. 


149 


it  has  attained  a  certain  magnitude,  gradually  increases,  at  the 
end  of  its  decrease  it  is  called  a  minimum. 

Thus,  if  a  line  F'R',  moving  from  A 
along  AB,  gradually  decreases  until  it 
comes  into  the  position  PR,  and  after 
that  gradually  increases,  the  line  is  said 
to  be  a  minimum,  or  at  its  least  value, 
when  it  comes  into  the  position  PR.  R  R  R" 

(205.)  If  u  be  a  function  of  x,  and  if  we  represent  by  u  the 
value  which  u  assumes  when  x  is  decreased  by  an  indefinitely 
small  quantity,  and  by  u"  the  value  which  u  assumes  when  x 
is  increased  by  an  indefinitely  small  quantity  ;  then,  if  u  be 
greater  than  both  u'  and  u",  it  will  be  a  maxiiium  ;  if  u  be 
less  than  both  u'  and  u",  it  will  be  a  minimum. 

Hence  the  maximum  value  of  a  variable  function  exceeds  those 
values  which  immediately  precede  and  follow  it,  and  the  mini- 
mum value  of  a  variable  function  is  less  than  those  values  which 
immediately  precede  and  follow  it. 

(206.)  We  have  seen,  Art.  201,  that  if?/  represents  the  or- 
dinate, and  x  the  abscissa  of  any  curve,  the  tangent  of  the  an- 
gle which  the  tangent  line  forms  with  the  axis  of  abscissas,  will 
be  represented  by 

dy 
dx' 

If  PR  becomes  a  maximum,  the  tan- 
gent TP,  being  then  parallel  with  the  axis   T- 
of  abscissas,  makes  no  angle  with   this 
axis,  and  we  have 
dy 
dx 
If  PR  becomes  a  minimum,  the  tangent 
TP,  being  then  parallel  with  the  axis  of 
abscissas,  makes  no  angle  with  this  axis, 
and  we  have 

dy 


R 


dx 


=  0. 


R 


dy 


Thus,  the  equation  -f-=0  simply  expresses  the  condition  that 
the  tangent  at  P  is  parallel  with  the  axis  of  abscissas ;  and, 


150  Differential   Calculus. 

consequently,  the  ordinate  to  that  point  of  the  curve  may  be 

either  at  its  maximum  or  minimum  value. 

(207.)  In  order,  therefore,  to  determine  whether  a  function 

has  a  maximum  or  a  minimum  value,  we  make  its  first  differ 

du 
ential  coefficient,  — ,  equal  to  zero,  and  find  the  value  of  a:  ir 

this  equation.  Represent  this  root  by  a.  Then  substitute  suc- 
cessively for  x  in  the  given  function,  a+A  and  a  —  h.  If  both 
he  results  are  less  than  the  one  obtained  by  substituting  a,  this 
value  will  be  a  maximum  ;  if  both  results  are  greater,  this  value 
will  be  a  minimum. 

Ex.  1.  Find  the  value  of  £  which  will  render  u  a  maximum 

in  the  equation 

u—\0x  — x\ 

Differentiating,  we  obtain 

du 

-r=10-2x. 

ax 

Putting  this  differential  coefficient  equal  to  zero,  we  have 

10-2.r=0. 

Whence  x=5. 

Let  us  now  substitute  for  x  in  the  given  function  5,  5—1 
and  5  +  1  successively. 

Substituting  4  for  x,  we  have  u'  =40  —  16  =  24. 

5    "  x,         "         u  =50-25=25. 

"  6    "  x,         "        w"  =  60-36  =  24. 

The  results  of  the  substitution  of  5—1  and  5  +  1  for  x  are 
both  less  than  that  obtained  by  substituting  5.  Hence  the 
function  u  is  a  maximum  when  x=5. 

Ex.  2.  Find  the  value  of  x  which  will  render  u  a  minimum 

in  the  equation 

M=.r2- 16^  +  70. 

Differentiating,  we  obtain 

du 

-r-=2x-16. 

dx 

Putting  this  equal  to  zero,  we  have 

2x-16=0. 
Whence  x=8. 


Maxima    ant    Minima    or    Functions.  151 

Let  us  now  substitute  for  x  in  the  given  function  8,  8—1 
and  8+1  successively. 

Substituting  7  for  x,  u'  =19-112  +  70  =  7. 

8  "    x,  u  =64-128  +  70  =  6. 

9  "    x,  u"=81  — 144+70=7. 

The  results  of  the  substitution  of  8  —  1  and  8  +  1  for  x  are 
both  greater  than  that  obtained  by  substituting  8.  Hence  the 
function  u  is  a  minimum  when  x=8. 

(208.)  A  general  method  of  determining  maxima  and  mini- 
ma of  functions  of  a  single  variable,  may  be  deduced  from 
Taylor's  theorem.  Art.  195. 

Suppose  we  have  u=F(x), 

and  let  the  variable  x  be  first  increased  by  h,  and  then  dimin- 
ished by  h  ;  and  let 

u'=F(x-h),  u"=F(x+h) ; 
then,  by  Taylor's  theorem,  we  shall  have 
du       cPu  Ji2      (Tu    h* 

u~u=:    dxh+d?U+d?Y^^etc- 

du.     d2u  h2      cPu    h3 
dx       dx   1.2     dx   1.2.3 

Now,  in  order  that  u  may  be  a  maximum,  it  must  be  greater 
than  either  v!  or  u" ;  that  is,  the  second  members  of  both  the 
above  equations,  for  an  infinitely  small  value  of  h,  must  be  neg- 
ative ;  and  in  order  that  u  may  be  a  minimum,  it  must  be  less 
than  either  vf  or  u" ;  that  is,  the  second  members  of  the  above 
equations,  for  an  infinitely  small  value  of  h,  must  be  positive. 
Now  when  h  is  infinitely  small,  the  sum  of  each  series  in  the 
above  equations  will  have  the  same  sign  as  the  first  term,  be- 
cause the  first  term  will  be  greater  than  the  sum  of  all  the  suc- 
ceeding ones.  But  the  first  terms  have  contrary  signs  ;  hence 
the  function  u  can  have  neither  a  maximum  nor  a  minimum,  un- 
less the  first  term  of  each  series  be  zero,  which  requires  that 

-  =  0, 
dx 

and  the  roots  of  this  equation  will  give  all  the  values  of  x  which 
can  render  the  function  u  either  a  maximum  or  a  minimum. 

Having  made  the  first  differential  coefficient  equal  to  zero, 
the  sign  of  the  sum  of  each  series  will  be  the  same  as  the  sign 
of  the  second  differential  coefficient. 


152  Differential    Calculus. 

If  the  second  differential  coefficient  is  negative,  the  function 
is  a  maximum  ;  if  positive,  a  minimum. 

If  the  second  differential  coefficient  reduces  to  zero,  the  sierns 
of  the  series  will  again  be  opposite,  and  there  can  be  neither  a 
maximum  nor  a  minimum  unless  the  third  differential  coefficient 
reduces  to  zero,  in  which  case  the  sign  of  the  sum  of  each  series 
will  be  the  same  as  that  of  the  fourth  differential  coefficient. 

(209.)  Hence,  in  order  to  find  the  values  of  x  which  will  ren- 
der the  proposed  function  a  maximum  or  a  minimum,  we  have 
the  following 

Rule. 

Find  the  first  differential  coefficient  of  the  function ;  place  it 
equal  to  zero,  and  find  the  roots  of  the  equation. 

Substitute  each  of  these  roots  in  the  second  differential  coeffi- 
cient. Each  one  which  gives  a  negative  result  will,  when  substi- 
tuted in  the  function,  make  it  a  maximum,  and  each  which  gives 
a  positive  result  will  make  it  a  minimum. 

If  either  root  reduces  the  second  differential  coefficient  to  zero, 
substitute  in  the  third,  fourth,  etc.,  until  one  is  found  which  does 
not  reduce  to  zero.  If  the  differential  coefficient  which  does  not 
reduce  to  zero  be  of  an  odd  order,  this  root  will  not  render  the 
function  either  a  maximum  or  a  minimum.  But  if  it  be  of  an  even 
order  and  negative,  the  function  will  be  a  maximum;  if  positive, 
a  minimum. 

Ex.  1.  Find  the  values  of  a;  which  will  render  u  a  maximum 
or  a  minimum  in  the  equation 

z^=.t3-3.c2-24z+85. 
Differentiating,  we  obtain 

du 

-r  =  3x2-C)x—24. 

dx 

Placing  this  equal  to  zero,  we  have 

3.z2-6x-24=0, 

or  x*  —  2x—  8  =  0, 

the  roots  of  which  are  +  4  and  —2. 

The  second  differential  coefficient  is 

d'u 

-j-2=6x-6. 

ax' 

Substituting  4  for  x  in  the  second  differential  coefficient,  the 
result  is  4- IS,  which,  being  positive,  indicates  a  minimum  ;  sub- 


Maxima    and   Minima    of   Functions 


If)  3 


stituting  -2  for  x,  the  result  is  -18,  which,  being  negative,  in- 
dicates a  maximum. 

Hence  the  proposed  function  has  a  maximum  value  when 
x=  —  2,  and  a  minimum  value  when  x=4. 

This  result  may  be  illustrated  by  assuming  a  series  of  values 
for  x,  and  computing  the  corresponding  values  of  u.     Thus, 

If?— -4,  u=  69, 
x—— 3,  m=103, 

x=—  2 ,  u=113  ?n aximum. 

x—  —  l,  u=l05, 

x=     0,  u—  85, 

x=  +  l,  u=   59, 

a:=+2,  w=  33, 

x=  +  3,  u=   13, 

a:=+4,  u=     5  minimum. 

x=-{-5,  u=    15, 

z=  +  6,  M=   49. 

Thus  it  is  seen  that  the  value  of  the  function  increases,  while 
x  increases  from  —4  to  —2;  it  then  decreases  till  x=4,  and 
after  that  it  increases  again  uninterruptedly,  and  will  continue 
to  do  so  till  x  equals  infinity.  This  peculiarity  may  be  illus- 
trated by  a  figure. 

If  we  assume  the  different 
values  of  x  to  represent  the 
abscissas  of  a  curve,  and  erect 
ordinates  equal  to  the  corre- 
sponding values  of  u,  the  curve 
line  which  passes  through  the 
extremities  of  all  the  ordinates, 
will  be  of  the  form  represented 
in  the  annexed  figure,  where 
it  is  evident  that  the  ordinates 
attain  a  maximum  corresponding  to  the  abscissa  —2,  and  a 
minimum  corresponding  to  the  abscissa  4. 

Ex.  2.  Find  the  values  of  a:  which  will  render  u  a  maximum 
or  a  minimum  in  the  equation 

tt=x3-18a;2+96.r-20. 
Ans.  x=4  renders  the  function  a  maximum,  and  £=8  renders 
it  o  minimum. 


154 


J) 


1FFERKNTIAL 


Cal 


CULTS. 


Ex.  3.  Find  the  values  of  x  which  will  render  it  a  maximum 
or  a  minimum  in  the  equation 

w=x3-18x2  +  105.r. 

Ans.  This  tunction  has  a  maximum  value  when  x=5,  and  a 
minimum  value  when  x=l. 

Illustrate  these  results  by  a  figure,  as  in  the  preceding  ex- 
ample. 

Ex.  4.  Find  the  values  of  x  which  will  render  u  a  maximum 
or  a  minimum  in  the  equation 

u=xi-l  Gx3 + 88.r  - 1 92x  + 1 50. 
Ans.  This  function  has  a  maximum  value  when  x  =  4.  and  a 
minimum  value  when  x=2  or  G. 

If  we  assume  a  series  of  values  for  x,  we  shall  obtain  the 
corresponding  values  of  u  as  follows  : 
If  x=l,  u=   31, 

.t=2,  m=     6  minimum. 
x—3,  u=    15, 
x=4,  u=  22  maximum. 
x=5,  u—    15. 
x=6,  u=     6  minimum. 
x=l,  u=  31, 
£=8,  m=150. 
The  curve  representing  these 
values  has  the  form  represented 
in  the  annexed  figure,  where  two 
minima  are  seen  corresponding 
to  the  abscissas  2  and  6,  and  a 
maximum  corresponding  to  the 
abscissa  4. 

Ex.  5.  Find  the  values  of  x 
which  will  render  u  a  maximum 
or  a  minimum  in  the  equation 

700 
u = x*  -  25x4 + — -x3  - 1 000.ra  + 1 920.T  - 1 1 00. 
o 

Ans.  This  function  has  two  maximum  values  corresponding 
to  x=2  and  x=Q.  and  two  minimum  values  corresponding  to 
x=4  and  x=8. 

(210.)  The  following  principles  will  often  enable  us  to  abridge 
the  process  of  finding  maxima  and  minima: 


6     18 


Maxima    and   Minima   of   Functions. 


155 


1.  If  the  proposed  function  is  multiplied  or  divided  by  a  con- 
stant quantity,  the  same  values  of  a;  which  render  the  function 
a  maximum  or  a  minimum,  will  also  render  the  product  or 
quotient  a  maximum  or  minimum  ;  hence  a  constant  factor 
may  be  omitted. 

2.  Whatever  value  of  x  renders  a  function  a  maximum  or  a 
minimum,  must  obviously  render  its  square,  cube,  and  every 
other  power  a  maximum  or  a  minimum  ;  and  hence,  if  a  func- 
tion is  under  a  radical,  the  radical  may  be  omitted. 

In  the  solution  of  problems  of  maxima  and  minima,  we  must 
obtain  an  algebraic  expression  for  the  quantity  whose  maxi- 
mum or  minimum  state  is  required,  find  its  first  differential  co- 
efficient, and  place  this  equal  to  zero  ;  from  which  equation 
the  value  of  the  variable  x,  corresponding  to  a  maximum  or  a 
minimum,  will  be  obtained. 

(211.)  The  following  examples  will  illustrate  these  principles. 

Ex.  1.  It  is  required  to  find  the  maximum  rectangle  which 
can  be  inscribed  in  a  given  triangle. 

Let  b  represent  the  base  of  the  trian- 
gle ABC,  h  its  altitude,  and  x  the  alti- 
tude of  the  inscribed  rectangle.     Then, 
by  similar  triangles,  we  have 

CD  :   CG  : :  AB  :  EF, 
or  h   :  h-x  :  :    b    :  EF. 


Hence 


EF=S(A-*). 


Therefore  the  area  of  the  rectangle  is  equal  to  EFxGD, 


or 


j{hx-x*), 


which  is  to  be  a  maximum. 

But  since  j  is  a  constant,  the  quantity  hx—x*  will  also  be  a 

maximum,  Art.  210. 

du     , 

—=h-2x=0, 

ax 

h 


Hence 


or 


x=: 


Hence  the  altitude  of  the  rectangle  is  equal  to  half  the  alti 
tude  of  the  triangle. 


56 


Differential    Calcui'js. 


Ex.  2.  What  is  the  altitude  of  a  cylinder  inscribed  in  a  given 
right  cone  when  the  solidity  of  the  cylinder  is  a  maximum  ? 

EeT  a  represent  the  height  of  the  cone,  b 
the  radius  of  its  base,  and  x  the  altitude  of 
the  inscribed  cylinder.      Then,  by  similar 


triangles,  we  have 

AD:  BD 

a   :    b 


or 
Hence 


AE  :EF, 
a-x  :  EF. 


EF=-(a-x), 


Now  the  area  of  a  circle  whose  radius  is  R  is  ttR2  (Geom., 

Prop.  XIII.,  Cor.  3,  B.  VI.).     Hence  the  area  of  a  circle  whose 

radius  is  EF  is 

nb* 

— j-(a— a:)*. 

Multiplying  this  surface  by  DE,  the  height  of  the  cylinder, 
we  obtain  its  solidity, 

—x{a-x)\ 


which  is  to  be  a  maximum. 


7Tb' 


Neglecting  the  constant  factor  — -,  we  have 

2i—x(a— z)*=a*x— 2ax2-}-x\  a  maximum. 

Differentiating,  we  have 

du 

-7-=a'—4ax+ 3x~ = 0, 

where  x  may  equal  a  or  \a. 

The  second  differential  coefficient  is 

d2u 

~r^—  —  4a+6x. 

ax 

The  value  x~a  reduces  the  second  differential  coefficient  to 
a  positive  quantity,  indicating  a  minimum;  the  value  x=\a 
reduces  this  coefficient  to  a  negative  quantity,  indicating  a 
maximum  ;  that  is,  the  height  of  the  greatest  cylinder  is  one 
third  the  altitude  of  the  cone. 

Ex.  3.  What  is  the  altitude  of  the  maximum  rectangle  which 
f.an  be  inscribed  in  a  given  parabola  ? 


Maxima    and   Minima   of  Functions 

Put  AT)  =  a  and  AE=;c  ;  then,  by  A 

the  equation  of  the  parabola  y*=2px, 
we  have 

Hence  GE=  V2px,  and  GH=2  ^/2px. 

Therefore  the  area  of  GHK1  is 
2  V2px(a—x),  which  is  a  maximum, 
or      s/x{a—x)  is  a  maximum. 

Hence 

du 
dx 


And 
Hence 


u—  ax-— x~, 

—i         i 

=  \ax  2  —  #.r2=0. 


a 


x2 
or  a=3.r,  and  x—\a. 

Consequently  the  altitude  of  the  maximum  rectangle  is  two 
thirds  of  the  axis  of  the  parabola. 

There  is  a  parabola  whose  abscissa  is  9,  and  double  ordi- 
nate 16;  required  the  sides  of  the  greatest  rectangle  which 
can  be  inscribed  in  it.  Ans. 

Ex.  4.  What  is  the  length  of  the  axis  of  the  maximum  parab- 
ola which  can  be  cut  from  a  given  right  cone  ? 

Put  BC=a,AB-&, and  CE=x,  then  BE=a-x,FE=v/a.r-a;a, 
Geom.,  Prop.  XXII.,  Cor.,B.  IV.,  and  FG=2</^=P. 

By  similar  triangles  we  have 

hx 
a  :  b  : :  x  :  DE= — . 
a 

Hence  the  area  of  the  parabola,  Art.  G5,  is 

2  bx        . 

-.  —  .2Vax—x'2,  a  maximum. 

3  a 

Hence  we  find       x=%a, 

and  DE=— =%b; 

a 

that  is,  the  axis  of  the  maximum  parabola 
is  three  fourths  the  side  of  the  cone. 

Ex.  5.  Divide  a  into  two  parts  such  that  the  least  part,  mul- 
tiplied by  the  square  of  the  greatest,  may  be  a  maximum. 

a         2a 
Ans.  -  and  — . 

o  o 


158 


Differential   Calculus. 


Ex.  0.  Divide  a  into  two  parts  such  that  the  least,  multiplied 
by  the  cube  of  the  greatest,  may  be  a  maximum. 

Let  x  represent  the  greater  part,  then 

x*(a— x)=a  maximum, 
and  x=la. 

Ex.  7.  It  is  required  to  determine  the  dimensions  of  a  cylin- 
drical vessel  open  at  top,  which  has  the  least  surface  with  a 
given  capacity. 

Let  c  denote  the  capacity  of  the  vessel,  and  x  the  radius  of 
the  base;  the  area  of  the  base  will  be  represented  by  nx\ 

Hence  the  height  of  the  cylinder  equals  — . 

J  ^         -nx* 

c  2c 

The  convex  surface  of  the  cylinder  is  — -X2nx=— . 

J  TTX*  X 

Adding  to  this  the  area  of  the  base,  we  have 
2c 


— +7T.T  ,  a  minimum, 


lrom  which  we  obtain 


Substituting  this  value  of  x  in  the  expression  for  the  height, 

we  find  the  height  =V  -;  that  is,  the  altitude  of  the  cylinder 
is  equal  to  the  radius  of  the  base. 

Ex.  8.  Required  the  altitude  of  a  cone  inscribed  in  a  given 
sphere,  which  shall  render  the  convex  sur- 
face of  the  cone  a  maximum. 

Let  AC=2a,  and  AD=x,  then 
x  :  BD  : :  BD  :  2a-x. 
Whence  BD=  V2ax—x\ 

Also,        x  :  AB  : :  AB  :  2a. 
Whence  AB=  V2ax. 

The  convex  surface  of  the  cone  =n  V2ax— x2  V2ax, 

=  nV4a2x*—2ax3,a  maximum. 
Whence  x=±a; 

that  is,  the  altitude  of  the  cone  whose  convex  surface  is  a  max- 
imum, is  |  of  the  radius  of  the  sphere. 


Max 


IMA     AN 


d   Minima    of  Functions. 


159 


Ex.  9.  Required  the  greatest  right-angled   triangle  which 
can  be  constructed  upon  a  given  line. 

Let  a  represent  the  hypothenuse  AB,  and 
v  one  of  the  sides  of  the  triangle,  the  other 
side  will  be  y/a'—x\  and  the  surface  of  the 

x    i — i a 

triangle  will  be  -Va—x  . 

Whence  2x*=a\ 

Therefore  the  two  sides  of  the  required  triangle  are  equal  to 
each  other. 

Ex.  10.  Required  the  least  triangle  which  can  be  formed  by 
the  radii  produced,  and  a  tangent  line  to  the  quadrant  of  a 
given  circle. 

Let  ABC  be  the  required  triangle,  and  draw    c 
AD  from  the  right  angle  perpendicular  to  the 
hypothenuse.     The  area  of  the  triangle  ABC 
is'  equal  to  iADxBC,  which  will  be  a  mini- 
mum when.BC  is  a  minimum,  because  AD  is 

...  A  B 

a  constant  quantity. 

Let  AD  =  R,  and  BD=z;  then,  Georn.,  Prop.  XXII.,  B.  IV., 
BD  :  AD  : :  AD  :  DC, 

BD      x 


or 


Hence 


R2 

BC=z-f— . 

x 


Therefore  x=~R,  and  DC=R  ;  that  is,  the  two  sides  of  the 
required  triangle  are  equal  to  each  other. 

Ex.  11.  A  right-angled  triangle  is  to  contain  a  given  area; 
required  the  base  and  oerpendicular  so  that  their  sum  may  be 
the  least  possible. 

Ans. 

Ex.  12.  Required  the  least  square  which  can  be  inscribed  in 
a  given  square. 

Ans.  Each  angle  of  the  required  square  is  on  the  middle  of 
a  side  of  the  given  square. 

Ex.  13.  Required  the  sides  of  the  maximum  rectangle  in 

Bcribed  in  a  given  circle. 

Ans.  Each  is  equal  to  R-/2. 


180  Differential   Calculus. 

Ex.  14.  Required  the  maximum  cone  which  can  be  in- 
scribed in  a  given  sphere. 

Ans. 

Ex.  15.  It  is  required  to  determine  the  dimensions  of  a  cyl- 
inder which  shall  contain  a  cubic  foot,  and  have  the  least  pos- 
sible surface,  including  both  ends. 

Ans. 

Ex.  1G.  A  carpenter  has  a  tapering  tree  of  valuable  wood, 

he  diameter  of  the  larger  end  being  three  feet,  and  that  of  the 

smaller  end  a  foot  and  a  half,  and  the  length  20  feet;  and  he 

wishes  to  cut  the  largest  possible  cylinder  out  of  it ;  required 

the  length  and  diameter  of  the  cylinder  ? 

Ans. 

Ex.  17.  A  cabinet-maker  has  a  mahogany  board,  the  breadth 
at  one  end  being  4  feet,  and  at  the  other  2,  and  its  length  10 
feet ;  and  he  wishes  to  cut  the  largest  possible  rectangular 
table  out  of  it.  At  what  distance  from  the  narrow  end  must 
it  be  cut  ? 

Ans. 


SECTION    IV. 

OF  TRANSCENDENTAL  FUNCTIONS. 

(212.)  An  algebraic  function  is  one  in  which  the  relation  l»e- 
ween  the  function  and  variable  can  be  expressed  by  the  or- 
dinary operations  of  algebra,  as  in  the  functions  hitherto  con- 
sidered. 

A  transcendental  function  is  one  in  which  the  relation  bt 
tween  the  function  and  variable  can  not  be  expressed  by  the 
ordinary  operations  of  algebra  ;  as, 

z^=sin.  x,  w=tang,  x,  w=sec.  x,  etc., 
which  are  called  circular  functions, 
or  u=]og.  x,  u=a*, 

which  are  called  logarithmic  or  exponential  functions. 

Proposition  I. — Theorem. 

(213.)  The  differential  of  a  constant  quantity  raised  to  a 
power  denoted  by  a  variable  exponent,  is  equal  to  the  power 
multiplied  by  the  Naperian  logarithm  of  the  root,  into  the  differ- 
ential of  the  exponent. 

Let  us  take  the  exponential  function 

u=af, 
and  give  to  x  an  increment  //,  we  shall  have 

u'=a*+h=axah. 
Therefore  u'-u=axah-a*=ax(a*-l).  (1) 

In  order  that  ah  may  be  developed  into  a  series  by  the  bi 
nomial  theorem,  let  us  assume 

a=l+b, 
we  shall  then  have 

7  7  2  7,3 

ah=(l+b)h=l+hj+h{h-l)— +/i(/i-l)(A-2)— +,etc 

The  second  member  of  this  equation  consists  of  a  series  oi 

L 


162  Differential   Calculus. 

terms  involving  the  first  power  of  h,  together  with  terms  in- 
volving the  higher  powers.     It  may,  therefore,  be  written 

ah=(l+i)h=l  +  ( -——+——,  etc.)/i+terms  involving  higher 

powers  of  h. 

Let  us  put  k  for  the  expression 

b    ¥    b3 

l-2+3~'etC-' 

and  we  shall  have 

ah—  l=kh+  terms  involving  A\  etc. 

Substituting  this  value  in  equation  (1),  it  becomes 

u'  —  u—a*kh+  terms  in  A2,  etc., 

u1  —  u 
or  — -. — =axk+  terms  involving  h,  etc., 

which  expresses  the  ratio  of  the  increment  of  the  function  to 
that  of  the  variable,  and  we  must  find  the  limit  of  this  ratio  by 
making  the  increment  equal  to  zero,  Art.  174;  the  result  will 
be  the  differential  coefficient.     Hence 

du    da* 

dx     dx 

The  symbol  k  represents  a  constant  quantity  depending  on 

a,  and  its  value  may  be  found  by  Maclaurin's  theorem.     We 

have  found 

da* 

-T-=a*k. 

dx 


fdcf\ 
\dx) 


Hence  d[  —  )  =da*k=a*kidx. 

cFa* 
Therefore  -^-r=a*k2. 

dx* 

d3ax 
Also,  -T-r=a*k  ,  etc. 

dx3 

If  in  the  function  u—a*,  and  the  successive  differential  co- 
efficients thus  found,  we  make  x=0,  we  shall  obtain 

Hence,  by  substitution,  Art.  191, 

fiX       ft  X  ri  X 

a*=l  -1 1 1 K  etc. 

J       1.2      1.2.3     ' 


Transcendental  Functions.  163 

If  we  now  make  x=T,  we  shall  have 

The  sum  of  this  series  is  2.718282,  which  is  the  base  of  the 

Naperian  system  of  logarithms.      Representing  it  by  e,  we 

shall  have 

i_ 

ak=e, 

or  a  =ek. 

Hence  the  constant  quantity  k  is  the  Naperian  logarithm  of  a, 

which  we  will  denote  by  log.'  a.     Hence 

da*  .       , 

-^  log/  a, 

or  daK=a%.  log.'  a.dx. 

Proposition  II. — -Theorem. 

(214.)  The  differential  of  the  logarithm  of  a  quantity  is  equal 
to  the  modulus  of  the  system,  into  the  differential  of  the  quantity 
divided  by  the  quantity  itself. 

According  to  the  preceding  Proposition, 
da*=a*  log.'  a.dx. 

Put  u=ax,  and  we  find 

du 

dx=—. — . 

u  log.  a 

If  a  be  the  base  of  a  system  of  logarithms,  then  x  is  the  log- 
arithm of  u  in  that  system,  and  ; — ,  Algebra,  Art.  349,  ia 

J  log.'  a        ° 

the  modulus  of  the  system,  which  we  will  represent  by  M 

du 
Hence  dx=d.  log.  u=m. — . 

(215.)  Cor.  For  the  Naperian  system  of  logarithms  M=l 
and  the  preceding  expression  becomes 

du 

d.  log.'  u= — ; 

°  u 

that  is,  the  differential  of  the  Naperian  logarithm  of  a  quantity 
is  equal  to  the  differential  of  the  quantity  divided  by  the  quan- 
tity itself. 


164  Differential   Calculus. 

Ex.  1.  Required  the  differential  of  the  common    ogarithm 

of  4825. 

From  Art.  214  it  appears  that  if  we  divide  the  modulus  ot 
the  system  of  logarithms  by  4825  (regarding  its  differential  as 
unity),  we  shall  obtain  the  differential  of  the  logarithm  of  4825. 
This  division  may  be  performed  arithmetically,  or  by  the  use 
of  logarithms,  thus : 
The  modulus  of  the  common  system  is  .434294,  whose 

logarithm  is 9.637784. 

The  logarithm  of  4825  is 3.683497. 

The  difference  is 5.954287. 

The  number  corresponding  to  this  logarithm  is 
.000090, 
which  is  the  difference  between   the  logarithm  of  4825  and 
that  of  4826,  as  is  seen  in  the  Table  of  Logarithms,  p.  11. 

Ex.  2.  Required  the  differential  of  the  common  logarithm 
0f  9651.  Ans.  .000045. 

Ex.  3.  Required  the  differential  of  the  common  logarithm 
0f  5791.  Ans.  .000075. 

Ex.  4.  Required  the  differential  of  the  common  logarithm 
of  3810.  Ans.  .000114. 

(216.)  By  combining  the  preceding  theorems  with  those  be- 
fore given,  we  may  differentiate  complex,  exponential,  and 
logarithmic  functions. 

In  the  following  examples,  Naperian  logarithms  are  sup- 
posed to  be  employed.  If  the  logarithms  are  taken  in  any 
other  system,  we  have  merely  to  multiply  the  results  by  the 
modulus  of  that  system. 

i        a+x  ,         . 

Ex.  1.  Differentiate  the  function  w=log.  ■,  by  the  rule 


a—x 


for  fractions  and  that  for  logarithms. 


Ex.  2.  Differentiate  the  function  u  =  \og. 


2adx 
Ans.  — = ; 


Va'+x* 

cfdx 

Ans.        ,  — jr. 
x(a  +x  ) 


Differentiation   of   Circular   1'unctions.   1 65 

Ex.  3.  Differentiate  the  function  u=(ax  +  l)~. 

Ans.  2ax(ax+l)  log.  a.dx. 

a*— I 

Ex.  4.  Differentiate  the  function  u— ,  bv  the  rule  for 

a  +1 

fractions  and  that  for  exponential  functions,  and  find  the  nu- 
merical rate  of  increase  of  the  function  when  a=10,  and  x 
becomes  2. 

2a*  log.  a.dx 
Ans-  — /  ,Ttx« — =  0.045d.c. 
(a  +1) 

The  Naperian  logarithm  of  10  is  2.302.     See  Alg.,  Art.  348. 
Ex.  5.  Differentiate  the  function 

u=y% 
in  which  y  and  x  are  both  variables. 

H  we  take  the  logarithm  of  each  member  of  this  equation, 
eve  shall  have 

log.  u=x  log.  y. 

tt  du     ,  ,        dy 

Hence,  Art.  215,         — =  log.  ydx+x—, 

u  °  J  y 

or  du=u  log.  ydx+ux—  ; 

y 

or,  by  substituting  for  u  its  value,  we  have 

du=dyx=yx  log.  ydx-\-xyx~ldy, 

which  is  evidently  the  sum  of  the  differentials  which  arise  by 

differentiating,  first  under  the  supposition  that  x  varies  and  y 

emains  constant,  and  then  under  the  supposition  that  y  varies 

and  x  remains  constant. 

ax 
Ex.  G.  Differentiate  the  function  u=—  or 

xx        \x 


-)'■ 

xJ 
Ans.    (J J  (log.  ™l)<k. 


DIFFERENTIATION  OF  CIRCULAR  FUNCTIONS. 

Proposition  III. — Theorem. 

(217.)  An  arc  of  a  circle  not  exceeding  a  quadrant,  is  greatei 
than  its  sine,  and  less  than  its  tangent. 

Let  AB  be  an  arc  of  a  circle  whose  sine  is  BE  and  tangent 
AD.  Take  AB'  equal  to  AB,  and  draw  the  sine  B'E,  and  the 
tanjrent  D'A. 


i  66  Differential   Calculus. 

The  chord  BB',  being  a  straight  line,  is  B  D 

shorter  than  the  arc  BAB' ;  therefore  the 
6ine  BE,  half  of  BB',  is  less  than  the  arc  B  A, 
half  of  the  arc  BAB'.  Therefore  the  sine 
is  less  than  the  arc. 

Again,  the  area  of  the   sector  ABC   is 
measured  by 

lACXarc  AB. 

Also,  the  area  of  the  triangle  ADC  is  measured  by 
lACxAD. 

But  the  sector  ABC  is  less  than  the  triangle  ADC,  being  con- 
tained within  it ;  hence 

lACxarc  AB<iACxAD. 
Consequently  arcAB<AD; 

that  is,  the  arc  is  less  than  the  tangent. 

(218.)   Cor.  1.  The  limit  of  the  ratio  of  the  sine  to  the  art. is 

unity ;  for  when  the  arc  h  represented  by  AB  becomes  zco, 

the  ratio  of  the  sine  to  the  tangent  is  unity.     Since,  by  Trig. 

Art.  28, 

sin.      cos. 

tang.      R 

But  the  cosine  of  0  is  equal  to  radius ;  hence,  when  ^=0, 

sin.  h 

tang,  h 

and  since  the  arc  is  always  comprised  between  the  sine  and 

the  tangent,  we  must  have,  when  we  pass  to  the  limit, 

sin.  h 

— =1- 

Cor.  2.  Since  the  chord  of  any  finite  arc  is  less  than  the  arc, 
but  greater  than  the  sine,  and  we  have  found  that  the  limiting 
ratio  of  the  sine  to  the  arc  is  unity,  the  limiting  ratio  of  the 
chord  to  the  arc  is  unity. 

Proposition  IV. — Theorem. 

(219.)  The  differential  of  the  sine  of  an  arc  is  equal  to  the  co- 
sine of  the  arc,  into  the  differential  of  the  arc,  divided  by  radius. 

Let  M=sin.  x. 

If  we  increase  x  by  h,  then 


Differentiation   of   Circular  Functions.    167 

&' =  sin.  (x-t/i), 
and  u'  —  u~ sin.  (x+h)~ sin.  x.  (1) 

But  by  Trig.,  Art.  75, 

sin.  A- sin.  B=^  sin.  |(A-B)  cos.  i(A+B).       (2) 

Put  A=x+h,  and  B=.r,  equation  (2)  becomes 

2 

sin.  (xi-h)  —  sin.  #=tT  sin.  \h  cos.  (x+\li). 

Hence  equation  (1)  becomes 

U'  —  U  =  -o    Sm'    V1   C0S'   (^+2^)' 

Dividing  both  members  by  h,  and  both  terms  of  the  fraction 
in  the  second  member  by  2,  we  have 

u'  —  u    sin.  \h    cos.  (x+yi) 
~h~=~Jh~X  R  ' 

which  expresses  the  ratio  of  the  increment  of  the  function  to 
that  of  the  variable,  and  we  must  find  the  limit  of  this  ratio  by 
making  the  increment  equal  to  zero,  Art.  174.  But  in  this 
case,  according  to  the  last  proposition,  Cor.  1, 

sin.  ±h 


Hence 


du    cos.  x 


dx        R    ' 

cos.  xdx 


or  du=d  sin.  x—       „ 

Ex.  1.  Required  the  differential  of  the  sine  of  30°,  the  differ 
ences  being  taken  for  single  minutes. 

The  differential  of  #,  which  is  1',  must  be  taken  in  parts  of 
radius,  which,  on  p.  150  of  the  Tables,  is  found  to  be  .0002909. 

Its  logarithm  is 6.463726. 

The  logarithmic  cosine  of  30°  is 9.937531. 

Their  sum  is 6.401257. 

The  natural  number  corresponding  to  this  logarithm  is 
0.000252, 
which,  on  p.  122  of  the  Tables,  is  seen  to  be  the  difference  be- 
tween the  natural  sine  of  30°  and  the  sine  of  30°  1'. 
Ex.  2.  Required  the  differential  of  the  sine  oT  10°  31'. 

Ans.  0.000286. 

V 


1(J8  Differential   Calculus. 

Ex.  3.  Required  the  differential  of  the  sine  of  60°  46'. 

Arts.  0.000142. 
Ex.  4.  Required  the  differential  of  the  sine  of  80°  41'. 

Ans.  0.000047. 

Proposition  V. — Theorem. 
(220.)   The  differential  of  the  cosine  of  an  arc  is  negative,  and 
is  equal  to  the  sine  of  the  arc  into  the  differential  of  the  arc, 
divided  by  radius. 

Let  w  =  cos.  x. 

Tnen  du=dcos.x=ds'm.(90°—x).  (1) 

But,  by  the  last  Proposition, 

v  J  R 

But  cos.  (90o-z)=sin.ar, 

and  d(90°-x)  =  -dx. 

Hence,  by  substitution,  equation  (1)  becomes 

,  sin.  xdx 

d  cos.  x— s . 

R 

Cor.  Since  the  versed  sine  of  an  arc,  less  than  ninety  cie 
grees,  is  equal  to  radius  minus  the  cosine,  we  have 

j  ____„  i    •  7/r>  .     sin.  xdx 

d.  versed  sin.  x~d(R  —  cos.  x)— - . 

R 

Ex.  1.  Required  the  differential  of  the  cosine  of  65°  10'. 

Ans.  -0.000264. 
Ex.  2.  Required  the  differential  of  the  cosine  of  5°  31'. 

Ans.  —0.000028. 
As  the  arc  increases,  its  cosine  diminishes;  hence  its  differ 
ential  is  negative. 

Proposition  VI. — Theorem. 
(221 .)    The  differential  of  the  tangent  of  an  arc  is  equal  to  the 
square  of  radius,  into  the  differential  of  the  arc,  divided  by  Uic 
square  of  the  cosine  of  the  arc. 


Let 


tt=tanir.  x. 


Q- R  sin.  x 

^nce  tang.  x= ,  we  have,  Art.  184, 

cos.  x 


Differentiation   of  Circular   Functions.    169 
R  cos.  xd  sin.  x—K  sin.  xd  cos.  x 


d. 

tang. 

X- 

cos.2  X 
(cos.5  .r  +  sin.2  x)dx 
cos.2  X 

But 

cos.2  x+sin.2  &=R2. 

Hence 

R2e?x 

d.  tang.  £= 5 — . 

Ex.  1.  Required  the  differential  of  the  tangent  of  45°. 

4ns.  0.00058. 

Ex.  2.  Required  the  differential  of  the  tangent  of  64°  14 

Ans.  0.00154. 

Proposition  VII. — Theorem. 

(222.)  The  differential  of  the  cotangent  of  alt  arc  is  negative, 
and  is  equal  to  the  square  of  radius  into  the  differential  of  the 
arc,  divided  by  the  square  of  the  sine  of  the  arc. 

Let  u=coi.  x. 

du=d  cot.  x=d  tang.  (90°-x).  (1) 

But  by  the  last  Proposition, 

v      Rad(90°-x) 
d^g.(^-x)  =  coJ{90O_^y 

Also,  d(90°-x)  =  —  dx, 

and  cos.2  (90°— x)  =  sin.2  x. 

Hence,  by  substitution,  equation  (1)  becomes 

Wdx 


d.  cot.  x= 


sin.  x 


Ex.  1.  Required  the  differential  of  ihe  cotangent  of  35°  6' 

Ans.  -0.00088. 

Ex.  2.  Required  the  differential  of  the  cotangent  of  21°  35'. 

Ans.  -0.00215. 

The  preceding  are  the  differentials  of  the  natural  sines,  tan- 
gents, etc.  The  differentials  of  the  logarithmic  sines  and  tan- 
gents may  be  found  by  combining  Proposition  II.  with  the  pre- 
ceding. 


170  Differential    Calculus. 

Proposition  VIII. — Theorem. 

(223.)  The  differential  of  the  logarithmic  sine  of  an  arc,  is 
equal  to  the  modulus  of  the  system  into  the  differential  of  the  arc. 
divided  by  the  tangent  of  the  arc. 

By  Proposition  II., 

Md  sin.  x    M  cos.  xdx 

d  JOg.  Sin.  X  — : = — pi — : . 

sin.  x  R  sin.  x 

R  sin   x 

But  Trig.,  Art.  28,    tang.  x= '—. 

b  '         &  C0S-  Xm 

Mdx 

Hence  d  log.  sin.  x= . 

tang,  x 

Ex.  1.  Required  the  differential  of  the  logarithmic  sine  of 

10'  30",  the  difference  being  taken  for  single  seconds. 

The  differential  of  x,  which  is  1",  must  be  taken  in  parts  of 

radius,  which,  on  p.  150  of  Tables,  is  found  to  be  .00000485. 

Its  logarithm  is    ...     .     4.G85575. 

The  modulus  M  log.      .     .     9.637784. 

Mdx 4.323359. 

tang.  10'  30" 7.484917. 

0.000689=0.838442. 

Therefore  0.000689  is  the  difference  between  the  logarithmic 

sine  of  10'  30"  and.  10'  31",  which  corresponds  with  p.  22  of 

the  Tables. 

Ex.  2.  Required  the  differential  of  the  log.  sine  of  4°  28'. 

Ans    0.000027. 

Proposition  IX. — Theorem. 

(224.)  The  differential  of  the  logarithmic  cosine  of  an  arc  is 
negative,  and  is  equal  to  the  modulus  of  the  systeni  into  the  tan- 
gent of  the  arc  into  the  differential  of  the  arc,  divided  by  the 
square  of  radius. 

By  Proposition  II., 

Md  cos.  x        M  sin.  xdx 

a  log.  cos.  x= = p^ . 

cos.  x  R  t_os.  x 

,„      „,  .  sin.  x     tan<r.  x 

But  Trig.,  Art.  28,      = — §—. 

c  cos..?;         R 

rT                                                        M  tang,  xdx 
Hence  d  log.  cos.  x  = ^ . 


Differentiation    of    Circi  lar    Functions.  171 

Ex.  1.  Required  the  differential  of  the  log.  cosine  of  67°  30' 

Ans.  -0.000005. 

Ex.  2.  Required  the  differential  of  the  log.  cosine  of  89°  30 
30".  Ans.  - 0.000245. 

Proposition  X. — Theorem. 

(225.)  The  differential  of  the  logarithmic  tangent  or  cotan- 
gent of  an  arc,  is  equal  to  the  modulus  of  the  system,  into  radius, 
'nto  the  differential  of  the  arc,  divided  by  the  product  of  the  sine 
2nd  cosine  of  the  arc. 

By  Proposition  II., 

Md  tang,  x  MWdx 

d  lo2f.  tanff.  x— 


tang,  x        cos.2  x  ^ang.  x 

But  Trig.,  Art.  28, 

cos.  x  tang.  x=TL  sin.  x. 
M.Rrfa: 


Hence  d  log.  tabg.  x 

Again,     d  log.  cot.  x 


sin.  x  cos.  x 
Md  cot.  x  MR'dx 


cot.  x  sin.2  x  cot.  x 

But  sin.  x  cot.  a:=R  cos.  x. 

M.Rdx 

Hence  d  log.  cot.  x=  — — . 

sin.  x  cos.  x 

Hence  the  differentials  of  the  log.  tangent  and  cotangent  of 
any  arc,  differ  only  in  sign. 

It  may  also  be  easily  proved  that  the  differential  of  the  log- 
arithmic tangent  is  equal  to  the  arithmetical  sum  of  the  differ- 
entials of  the  sine  and  cosine  of  the  same  arc. 

Ex.  1.  Required  the  differential  of  the  log.  tangent  of  10'  30". 

Ans.  0.000G89. 

Ex.  2.  Required  the  differential  of  the  log.  tangent  of  10°  16'. 

Ans.  0.000012. 

Ex.  3.  Required  the  differential  of  the  log.  tangent  of  89° 
4-  30".  Ans.  0.000130. 

(22G.)  We  have  found  the  differentials  of  the  sine,  cosine, 
etc.,  in  terms  of  the  arc  as  an  independent  variable.  It  is 
sometimes  more  convenient  to  regard  the  arc  as  the  function 


172  Differential   Calculus. 

and  the  sine,  cosine,  etc.,  as  the  variable.     Let  us  represent 
any  arc  by  z,  and  let  us  put 

y=sin.  z. 

By  Art.  219,  we  have  , 

cos.  zdz 

<¥=— r—  ; 

whence  dz= — .  (1) 

cos.  z 

But  cos.2  z  +  sin.2  z=R2; 


whence  cos.  z=  VR2  — sin.2  z=  VR'— y1. 

Substituting  this  value  in  equation  (1),  we  obtain 

Rdy 

dz=    —  J     , 

which  is  «the  differential  of  an  arc,  its  sine  being  regarded  as  the 
independent  variable. 

Let  us  put  y'=cos.  z. 

By  Art.  220,  we  have 

sin.  zdz 
dy'= ^— ; 

whence  dz  = — t-*—. 

sin.  z 


But  sin.  z  =  VR~  —  cos.2z=  VR2— y'\ 

Hence  tfz= — - — , 

VR2-*/'2 

which  is  the  differential  of  an  arc,  its  cosine  being  regarded  as 
the  independent  variable. 

Let  us  put  x=  versed  sine  z. 

By  Art.  220,  Cor.,  we  have 

sin.  zdz 


dx= 
wnence  dz= 


R 
Rdx 


But  sin.  z=V(2R-x)x,  Geom.,  Prop.  XXII.,  Cor.,  B.  IV., 
and,  consequently, 

Rdx 

dz= — , 

y/ZRx-x* 

which  is  the  differential  of  an  arc,  its  versed  sine  being  regard 

ec  as  the  independent  variable. 


Differentiation   of   Circular   Functions.   173 

Let  us  put  t=tang.  z. 

By  Art.  221,  we  have 

Radz 


dl 
hence  dz— 


COS.     2 

cos.2  zdt 


R2 

But  (Trig.,  Art.  28)     — ^ 


Hence 


sec.  z 
cos.2  z        R2  R2  Ra 


Ra        sec.2  z     R2  +  tang.2  z    R2-Wa* 
R"-dt 


Therefore  (/:=„ 

it  +r 

which  is  the  differential  of  an  arc  its  tangent  being  regarded 

as  the  independent  variable. 

(227.)  If  we  make  R  equal  to  unity,  these  formulas  become 

dy 

1.  dz=        J     , 

Vl-f 

where  y  represents  the  sine  of  the  arc  z. 

-dy' 

2.  dz=         y  --, 

Vl-y» 

where  y'  represents  the  cosine  of  the  arc  z. 

~     7  dx 

3.  dz=  =, 

V2x-x* 

where  x  represents  the  versed  sine  of  the  arc  z. 

dl 

where  t  represents  the  tangent  of  the  arc  z. 

Ex.  1.  If  two  bodies  start  together  from  the  extremity  of  the 
diameter  of  a  circle,  the  one  moving  uniformly  along  the  di- 
ameter at  the  rate  of  10  feet  per  second,  and  the  other  in  the 
circumference  with  a  variable  velocity  so  as  to  keep  it  always 
perpendicularly  above  the  former ;  what  is  its  velocity  in  the 
circumference  when  passing  the  sixtieth  degree  from  the  start- 
ing point,  supposing  the  diameter  of  the  circle  to  be  50  feet? 

20 
Ans.  — -  leet  per  second. 
v  3 

Ex.  2.  If  two  bodies  start  together  from  the  extremity  of  the 


174  Differential   Calculus. 

diameter  of  a  circle,  the  one  moving  uniformly  along  the  tan- 
gent at  the  rate  of  10  feet  per  second,  and  the  other  in  the  cir- 
cumference with  a  variable  velocity,  so  as  to  be  always  in  the 
straight  line  joining  the  first  body  with  the  center  of  the  cir- 
cle ;  what  is  its  velocity  when  passing  the  forty-fifth  degree 
from  the  starting  point,  the  diameter  of  the  circle  being  50  feet  ? 

Ans.  5  feet  per  second. 

(228.)  Maclaurin's  theorem  enables  us  to  develop  the  sine 
of  ;c,  cosine  of  x,  etc.,  in  terms  of  the  ascending  powers  of  a;. 

Ex.  I.  It  is  required  to  develop  sin.  x  into  a  series. 

Let  w=sin.  x,  and  R=unity. 

-r»      a    x   ^,^      du  d"u 

Oy  Art.  219,     —  =     cos.  x,  -r-^=  —  sm.  x, 
ax  dx' 

d3u  d*u 

T~3—~ cos-  x,  -r-i=  +  sin.  x,  etc. 
dx3  dx 

If  now  we  make  x=0,  we  shall  have 

/  %     r.     fdu\  fd-u\ 

(M)=0'  is)-  '•  \d?)=0> 


<Fu\  fdlu\ 

d?)=-h   WJ=0'etC' 


X  X 

Therefore,  Art.  191,  sin.  x=x 1- ,  etc. 

2.3     2.3.4.5 

Ex.  2.  It  is  required  to  develop  cos.  x  into  a  series. 


Let 

u  —  cos. . 

X. 

By  Art. 

220, 

du 
dx 

—  sin.  x, 

cT~u 
dx" 

—  —  COS. 

X, 

d3u 
dx3~ 

sin.  x, 

dlu 
dx' 

=      COS. 

X, 

etc, 

If  now  i 

we  make  x= 

■0,  we  shall 

have 

(u)  =  l, 

fdu\ 
\dxJ: 

=  0, 

Kdx'J 

1, 

(d3u\ 

=  0, 

\dx*J  = 

=1. 

lerefore. 

,  Art. 

191,  cos.  x=l- 

x*       xk 
2     2.3.4 

,  etc. 

These  series  for  small  values  of  x  converge  rapidly,  and  are 
very  convenient  for  computing  a  table  of  natural  sines  and 
cosines 


SECTION   V. 

APPLICATION  OF  THE  DIFFERENTIAL  CALCULUS  TO  THE  THEO- 
RY OF  CURVES. 

(229.)  If  we  differentiate  the  equation  of  a  line,  we  shall  ob- 
tain a  new  equation  which  expresses  the  relation  between  the 
differentials  of  the  co-ordinates  of  the  line.  This  equation  is 
called  the  differential  equation  of  the  line. 

If,  for  example,  we  take  the  equation  of  a  straight  line, 
y=ax+b.  (1) 

and  differentiate  it,  we  find 

I-  (*> 

a  result  which  is  the  same  for  all  values  of  b. 
Differentiating  equation  (2),  we  obtain 

g=0.  (3) 

This  last  equation  is  entirely  independent  of  the  values  of  a 
and  b,  arid  is  equally  applicable  to  every  straight  line  which 
can  be  drawn  in  the  plane  of  the  co-ordinate  axes.  It  is  called 
the  differential  equation  of  lines  of  the  first  order. 

(230.)  If  we  take  the  equation  of  the  circle 

x-+if  =  W,  (1) 

and  differentiate  it,  we  obtain 

2xdx+2i/dy=0, 

dy-     X  (9\ 

Tx—'y  (2) 

Equation  (2)  is  independent  of  the  value  of  the  radius  R,  and 
ncnce  it  belongs  equally  to  every  circle  referred  to  the  same 
co-ordinate  axes. 

If  we  take  the  equation  of  the  parabola 

y'=2px,  (1) 

and  differentiate  it,  we  find 

2ydy=2pdx  ; 


176  Differential   Calculus. 

whence  —-=—.  (2) 

dx     y  *  ' 

But  from  equation  (1),      p  =  ~- 

Hence  equation  (2)  becomes 

dy      y 

■   dlr~¥x  (3) 

This  equation  is  independent  of  the  value  of  the  parameter 
2p,  and  hence  it  belongs  equally  to  every  parabola  referred  to 
the  same  co-ordinate  axes. 

(231.)  If  we  take  the  general  equation  of  lines  of  the  second 
order,  which  is,  Art.  132, 

yt=mx+nx!>,  (1) 

and  differentiate  it,  we  obtain 

2ydy=m  dx + 2  nxdx.  (2) 

Differentiating  again,  regarding  dx  as  constant,  we  obtain 
2dif  +  2yd2y=2ndx\ 
or,  dividing  by  2.         dy2+  yd2y=  ndx\  (3) 

Eliminating  m  and  n  from  equations  (1),  (2),  and  (3),  we 
obtain 

y\lxn'+xidy- + yx^d'y — 2xydxdy = 0, 
which  is  the  general  differential  equation  of  lines  of  the  second- 
order. 

Hence  we  see  that  an  equation  may  be  freed  of  its  constants 
by  successive  differentiations ;  and  for  this  purpose  it  is  neces- 
sary to  differentiate  it  as  many  times  as  there  are  constants  to 
be  eliminated.  The  differential  equations  thus  obtained,  to 
gether  with  the  given  equation,  make  one  more  than  the  num- 
ber of  constants  to  be  eliminated,  and  hence  a  new  equation 
may  be  derived  which  will  be  freed  from  these  constants. 

The  differential  equation  which  is  obtained  after  the  con- 
stants are  eliminated,  belongs  to  a  species  of  lines,  one  of  which 
is  represented  by  the  given  equation. 

(232.)  We  have  seen,  Art.  201,  that  the  tangent  of  the  angle 
which  a  tangent  line  at  any  point  of  a  curve  makes  with  the 
axis  of  abscissas,  is  equal  to  the  first  differential  coefficient  of 
the  ordinate  of  the  curve.  We  are  enabled  from  this  princi- 
ple to  deduce  general  expressions  for  the  tangent  and  subtan- 
gent.  normal  and  subnormal  of  any  curve. 


DIFFERENTIAL     CALCULUS     OF    CtTRVES. 


177 


Proposition  I. — Theorem. 

(233.)  The  length  of  the  subtangent  to  any  point  of  a  curve 
referred  to  rectangular  co-ordinates,  is  equal  to  the  ordinate 
multiplied  by  the  differential  coefficient  of  the  abscissa. 

In  the  right-angled  triangle  PTR,  we  have, 
Trig.,  Art.  42, 

1  :  TR  : :  tang.  T  :  PR ; 
dy 


that 


is, 


1  :TR: 


dx  ' 


V- 


Hence  the  subtangent  TR=y 


dx 
dy 


R 


Proposition  II. — •Theorem. 

(234.)  The  length  of  the  tangent  to  any  point  of  a  curve  re- 
ferred to  rectangular  co-ordinates,  is  equal  to  the  square  root 
of  the  sum  of  the  squares  of  the  ordinate  and  subtangent. 

In  the  right-angled  triangle  PTR, 
TF=PR2  +  TR2; 

dx2 
that  is,  TF-=y-+y"- ' 

Hence  the  tangent 


dy- 


I  ,       „dx*  I      dx* 


dy9    *V      '  dy* 

Proposition  III. — Theorem. 

(235.)  The  length  of  the  subnormal  to  any  point  of  a  curve,  is 
equal  to  the  ordinate  multiplied  by  the  differential  coefficient  of 
the  ordinate. 

In  the  right-angled  triangle  PRN,  we 
have  the  proportion 

1  :  PR  : :  tang.  RPN  :  RN. 

But  the  angle  RPN  is  equal  to  PTR  ; 
hence       1  :  PR  :  :  tang.  PTR  :  RN ; 


that  is,      1  :    y 


dx 


ay 
Hence  the  subnormal  RN=y-f-. 

M 


178 


Differential    Calculus. 


Proposition  IV. — Theoreih. 

(236.)  The  length  of  the  normal  to  any  point  of  a  curve,  is 
equal  to  the  square  root  of  the  siun  of  the  squares  of  the  ordinate 
and  subnormal. 

In  the  right-angled  triangle  PRN, 
PN2  =  PR2+RN2: 


that  is,  PN2=t/2+^ 


dx1 


Hence  the  normal  PN 


=yy- 
=y\/i 


+ 


dx*  ' 


ay 

dx2' 


(237.)  To  apply  these  formulas  to  a  particular  curve,  we 

dx       d'li 

must  substitute  in  each  of  them  the  value  of  —  or  -f-,  obtained 

dy       dx  • 

by  differentiating  the  equation  of  the  curve.     The  results  ob- 
tained will  be  true  for  all  points  of  the  curve.     If  the  values 
are  required  for  a  given  point  of  the  curve,  we  must  substitute 
in  these  results  for  x  and  y  the  co-ordinates  of  the  given  point. 
Let  it  be  required  to  apply  these  formulas  to  lines  of  the 
second  order  whose  general  equation  is 
y*=mx-{-nx*. 
Differentiating,  we  have 

dy    m+2nx        ??i  +  2nx 
dx~     2y      ~2Vmx+nx*' 
Substituting  this  value  in  the  preceding  formulas,  we  find 

dx     2{mx-{-nxi) 


The  subtan^ent 


■y 


dy        m-\-2nx 


The  tangent  =  yV +^=\/ 


*f 


(?nx  +  nx 

mx  +  nx  +41 — 

\  m-\-2nx 


The  subnormal 


\'/ 


dy    m+2nx 
dx~       2      ' 


The  normal 


-V</+!/'c' 


dx" 


V mx+nx*  +l(m+2nxY 


(238.)  By  attributing  proper  values  to  m  and  n,  the  above 
formulas  will  be  applicable  to  each  of  the  conic  sections.  For 
the  parabola, n=0,  and  these  expressions  become 


Differential   Calculus   of  Curves.  179 

the  subtangent  =2x,  which  corresponds  with  Art.  53; 
the  tangent        =  Vmx+4x'2 ; 

771 

the  subnormal   =—  which  corresponds  with  Art.  56; 


the  normal         =  'y  mx-i . 

(239.)  In  the  case  of  the  ellipse,  these  expressions  assume  a 
simpler  form  when  the  origin  of  co-ordinates  is  placed  at  the 
center.     The  equation  then  becomes 

Ay+BV=A2B2,  (i) 

whence,  by  differentiating,  we  obtain 

A.~ydy+'Wxdx=0, 
dx__     A2y 

Hence,  from  Art.  233,  we  find  the  subtangent  of  the  ellipse 
equals 

Ay 

But  from  equation  (1),  we  have 

Hence  from  equation  (2),  we  find  the  subtangent  of  the  el- 
lipse equals 

_A2-xQ 

ir~ ' 

which  corresponds  with  Art.  78,  Cor.  2. 

Also,  from  Art.  235,  we  obtain  the  subnormal  of  the  ellipse 
equals 

dy_     B'x 
ydx~~  A3"' 
which  corresponds  with  Art.  80,  Cor.  1. 

(240.)  In  the  case  of  the  circle,  A  and  B  become  equal,  and 
we  find 

the  subtangent  = , 

°  x 

the  tangent        =  y y8+^=y  -JL=JLm 


180 


Differential   Calculus. 


the  subnormal  =  —x, 
the  normal 


=  Vy'1+x2=R, 
which  results  agree  with  well-known  principles  of  Geometry. 

Ex.  1.  If  the  parameter  of  a  parabola  be  4  inches  and  the 
abscissa  9  inches,  required  the  length  of  the  ordinate  and  sub- 
tangent.  Ans. 

Ex.  2.  If  the  major  axis  of  an  ellipse  be  30  inches  and  the 
minor  axis  16  inches,  required  the  length  of  the  subtangent 
corresponding  to  an  abscissa  of  10  inches  measured  from  the 
center.  Ans. 

Ex.  3.  If  the  major  axis  of  an  ellipse  be  6  inches  and  the 
minor  axis  4  inches,  required  the  length  of  the  subnormal  cor- 
responding to  an  abscissa  of  2  inches  measured  from  the  center. 

Ans. 

Ex.  4.  If  the  diameter  of  a  circle  be  10  feet,  what  is  the 
length  of  the  tangent  and  subtangent  corresponding  to  an  ab- 
scissa of  3  feet  measured  from  the  center? 

Ans. 

(241.)  Let  it  be  required  to  find  the  value  of  the  subtangent 
of  the  logarithmic  curve. 

Tf  we  differentiate  the  equation 

x=\og.  y,  Art.  141, 

and  represent  the  modulus  of  the  system  of  logarithms  by  M 
we  obtain,  Art.  214, 

Mdy 


dx= 


V 


or 


dx 


But  y-r-  is  the  expression  for  the  sub- 
tangent, Art.  233 ;  hence  the  subtan- 
gent of  the  logarithmic  curve  is  con- 
stant, and  equal  to  the  modulus  of  the 
system  in  which  the  logarithms  are 
taken. 

In  the  Naperian  system  M  equals 
unity,  and  hence  the  subtangent  AR 
will  be  equal  to  unity  or  AB. 


Differential   Calculus   of  Curves.  181 

(242.)  The  equation  of  a  straight  line  passing  through  a 
given  point,  Art.  18,  is 

y-y'=a(x-x'), 

where  a  denotes  the  tangent  of  the  angle  which  the  line  makes 
with  the  axis  of  abscissas. 

But  we  have  found,  Art.  201,  that  the  first  differential  co- 

dxi 
efficient  -j-  is  equal  to  the  tangent  of  the  angle  which  the  tan- 
gent line  to  a  curve  forms  with  the  axis  of  abscissas.     Hence 
the  equation  of  a  tangent  to  a  curve  at  a  point  whose  co-or- 
dinates are  x',  y',  is 

y-y'=%{x-x')-  (i) 

And  since  the  normal  is  perpendicular  to  the  tangent,  the 
equation  of  the  normal,  Art.  25,  must  be 

dr.' 

(243.)  When  it  is  required  to  find  the  equation  of  the  tan- 
gent line  to  any  curve,  we  must  differentiate  the  equation  of 

dy' 

the  curve,  and  find  the  value  of  — — ,  which  is  to  be  substituted 

dx' 

in  equation  (1). 
Ex.  1.  Let  it  be  required  to  find  the  equation  of  the  tangent 

line  to  a  circle. 

The  equation  of  the  circle  is 

and,  by  differentiating,  we  find 

dy'        x' 
dx'~~     if 
Substituting  this  value  in  equation  (1),  we  have  for  the  equa- 
tion of  a  tangent  line 

x' 

y-y'=--,(x-x')- 

Whence  yy,+xx,=>x»+y»='R\ 

which  corresponds  with  Art.  40. 
Ex.  2.  Find  the  equation  of  the  tangent  line  to  a  parabola. 


182 


Differential   Calculus. 


SUBTANGENT  AND  TANGENT  OF  POLAR  CURVES. 

(244.)  The  subtangenl  of  a  polar  curve 
is  a  line  drawn  from  the  pole  perpendicu- 
lar to  a  radius  vector,  and  limited  by  a 
tangent  drawn  through  the  extremity  of 
the  radius  vector. 

Thus,  if  MT  is  a  tangent  to  a  polar 
curve  at  the  point  M,  P  the  pole,  and  PM 
the  radius  vector,  then  PT,  drawn  per- 
pendicular to  PM,  is  the  subtangent. 


Proposition  V. — Theorem. 

(245.)  The  length  of  the  subtangent  to  a  polar  curve  is  equal 
to  the  square  of  the  radius  vector,  multiplied  by  the  differential 
coefficient  of  the  measuring  arc. 

Represent  the  radius  vector  PM  by 
r,  and  the  measuring  arc  ba  by  t  (the 
radius  Pa  of  the  measuring  circle  be- 
ing equal  to  unity).  Suppose  the  arc 
t  to  receive  a  small  increment  aa',  and 
through  a'  draw  the  radius  vector  PM'. 
With  the  radius  PM  describe  the  arc 
MN;  draw  the  chord  MN,  and  draw 
PT  parallel  to  MN. 

Now  aa'  is  the  increment  of  t,  and     \ 

M'N  is  the  increment  of?-;  and  in  or-  '' "' 

der  to  find  the  differential  coefficient  of  r  (t  being  considered 
the  independent  variable),  we  must  find  the  ratio  of  the  incre- 
ments of  t  and  r,  Art.  174,  and  determine  the  limit  of  this  ratio 
by  making  the  increment  of/  equal  to  zero. 

By  Geom.,  Prop.  XIII.,  Cor.  1,  B.  VI.,  we  have 

1  :  aa'  : :  PM  :  arc  MN. 

arc  MN 


Hence 


PM 


(1) 


Also,  the  similar  triangles  M'NM,  M'PT  furnish  the  pro- 
portion 

M'N  :  chord  MN  :  :  MP  :  PT. 


SUBTANGENT    AND    TANGENT    OF    PoLAR    CURVES.    183 

Whence  M'N= t^t '  ^ 

Consequently,  from  equations  (1)  and  (2), 
aa>  _    arc  MN  PT 

M7N~cIoTarMN  X  PM  X  PM" 
which  is  the  ratio  of  the  increments  of  t  and  r;  and  we  must 
now  find  the  limit  of  this  ratio  when  the  increment  of  t  is  made 
equal  to  zero. 

It  is  evident  that  the  ratio  of  the  arc  MN  to  the  chord  MN 

will  be  unity,  Art.  218,  Cor.  2  ;  also,  PT  will  be  the  subtangent, 

and  PM'  will  become  equal  to  PM,  which  is  represented  by  r. 

dt     PT 
Hence  ^-r, 

PT     r*dt 

which  is  the  value  of  the  subtangent. 

PT 

Cor.  The  tangent  of  the  angle  PMT  is  equal  to  p^|,  which 

rdt 
therefore  becomes  -7-.  which  represents  the  tangent  of  the  an- 
dr  l 

gle  which  the  tangent  line  makes  with  the  radius  vector. 

Pkoposition  VI. — Theorem. 

(246.)  The  length  of  the  tangent  to  a  polar  curve,  is  the  square 
root  of  the  sum  of  the  squares  of  the  subtangent  and  radius  vector. 

For  the  tangent  MT  is  equal  to  VMP2+PT2,  which  is  equal 

/      r*d¥ 
t0  rV1+"^T 

(247.)  It  is  required  to  apply  these  formulas  to  the  spirals. 
The  equation  of  the  spiral  of  Archimedes,  Art.  148,  is 

_  t 

dt 

Whence  -j-—2tt. 

dr 

■    Substituting  the  values  of  r  and  —  in  the  general  ejpressior, 

for  the  subtangent,  Art.  245,  we  have 

subtan"ent=—. 

0  2n 


184  Differential    Calculus. 

Tf  t—2rr,  that  is,  if  the  tangent  be  drawn  at  the  extremity 
of  the  arc  generated  in  one  revolution,  we  have 

the  subtangent=2n=the  circumference  of  the  measuring  circle. 

If  t=2?nT,  that  is,  if  the  tangent  be  drawn  at  the  extremity 
of  the  arc  generated  in  m  revolutions,  we  have 

subtangent— m  .  2m-x ; 
that  is,  the  subtangent  after  m  revolutions,  is  equal  to  m  times 
the  circumference  of  the  circle  described  with  the  radius  vector 
of  the  point  of  contact. 

248.)  The  equation  of  the  hyperbolic  spiral,  Art.  151,  is 

a 

Whence  -7-= — . 

dr        a 

Substituting  this  value  in  the  general  expression  for  the  sub- 
tangent,  we  have 

r-e 

subtangent= —  —  a  ; 

that  is,  in  the  hyperbolic  spiral  the  subtangent  is  constant. 
(249.)   The  equation  of  the  logarithmic  spiral,  Art.  155,  is 
t=\og.  r. 

Whence  dt— , 

r 

and  ~=M, 

dr 

which  represents  the  tangent  of  the  angle  which  the  tangent 
line  makes  with  the  radius  vector,  Prop.  V.,  Cor. ;  that  is,  the 
tangent  of  the  angle  which  the  tangent  line  makes  with  the  radius 
vector  is  constant,  and  is  equal  to  the  modulus  of  the  system  of 
logarithms  employed. 

DIFFERENTIALS  OF  AN  ARC,  AREA,  SURFACE,  AND  SOLID  OF 
REVOLUTION. 

Proposition  VII. — Theorem. 

(250.)  The  limit  of  the  ratio  of  the  chord  and  arc  of  any  curve 
ts  unity. 

Let  ADB  be  an  arc  of  any7  curve,  AB=c  the  chord,  and  let 


Differentials   of   an   Arc,   Area,   etc. 


185 


the  tangents  AC,  CB  be  drawn  at  the 
extremities  of  the  arc. 

It  is  evident  that  the   arc   ADB   is 
greater  than  the  chord  c,  but  less  than 
the  sum  of  the  two  tangents  a  and  b. 
By  Trigonometry,  Art.  49, 
a 

c     sin.  C 
a+b     sin.  A  +  sin.  B     sin 


sin.  A        .  b    sin.  B 

-  and  -=—. — ^. 

c     sin.  C 


Therefore 


•sin 


B 


sin.  (A  +  B) 


c  sin.  C 

By  Trigonometry,  Art.  76, 

sin.  A  +  sin.  B_cos.  A(A  —  B) 
sin.  (A+B)~-cos.  £(A  +  B)* 

a+b    cos.  i(A-B) 

Hence  —  ^o   i ,  A  ,r>\" 

c        cos.  £(A  +  r>) 

Conceive  now  the  points  A  and  B  to  approach  each  other,  and 
the  arc  ADB  to  decrease  continually,  the  angles  A  and  B  will 
manifestly  both  decrease,  and  they  may  become  less  than  any 
assignable  angle  whatever  ;  therefore  A— B  and  A+B  both  ap- 
proach continually  to  0  ;  and  cos.  i(A-B)  and  cos.  |(A+B)  ap- 
proach to  unity,  which  is  their  common  limit.  Hence  the  limit 
of  the  ratio  of  a+b  to  c  is  a  ratio  of  equality  ;  and  as  the  arc  ADB 
can  not  be  greater  than  a+b,  nor  less  than  c,  much  more  is 
the  limit  of  the  ratio  of  the  arc  to  the  chord,  a  ratio  of  equality. 

Proposition  VIII. — Theorem. 

(251.)  The  differential  of  the  arc  of  a  curve  referred  to  rect- 
angular co-ordinates,  is  equal  to  the  square  root  of  the  sum  oj 
the  squares  of  the  differentials  of  the  co-ordinates. 

We  have  found,  Art.  250,  that  the  limit 
of  the  ratio  of  the  chord  and  arc  of  a  curve 
is  unity  ;  hence  the  differential  of  an  arc 
is  equal  to  the  differential  of  its  chord. 

Let   x   represent    any    abscissa    of  a 
curve,  AR  for  example,  and  y  the  cor- 
responding  ordinate  PR.     If  now   we 
give  to  x  any  arbitrary  increment  h,  and  make  RR'  =  />,  the 
value  of?/  will  become  equal  to  P'R',  which  we  will  represent 


l8G  Differential   Calculus. 

by  y'.     If  we  draw  PD  parallel  to  the  axis  AR',  we  snail  have 

the  chord  PF=  VPD2 +FD,J=  v7*2+P'D2. 

^     ^.  ^,  dy,     d\i  If        ,  ...... 

But  r'\J=y'— pj^Hj^-  +  other  terms  involving  highei 

powers  of  A,  Art.  195. 

Substituting  this  value  of  P'D  in  the  expression  for  the  chord. 
we  have 


dx* 


V 


dy' 

+^+'  etC- 


Therefore  — -=v/  1  +^+,  etc., 

a        v         fife 

which  expresses  the  ratio  of  the  increment  of  the  function  to 

that  of  the  variable,  and  we  must  find  the  limit  of  this  ratio  by 

making  the  increment  equal  to  zero,  Art.  174. 

In  this  case  the  chord  becomes  equal  to  the  arc,  which  we 

will  represent  by  z,  and  the  terms  omitted  in  the  second  mfem 

ber  of  the  equation  containing  h  disappear  ;  hence 

dz       J      d^_ 

dx      *         dx* 


and,  multiplying  by  dx,  dz  =  vdz'+dy*. 

(252.)  To  determine  the  differential  of  the  arc  of  a  circle, 
take  the  equation 

xdx 

whence  xdx+ydy=0,  or  dy— , 


x*dx*     dx 


and  dz=\/dx*+ — ^=—  Vz*+y°. 

v  y      y      __ 

But  Vxi+y2=R,  and  y=  VR'2-x\ 

Hence  dz=    -    "— .      See  Art.  226. 

VR'-x' 

Proposition  IX. — Theorem. 

(253.)  The  differential  of  the  area  of  a  segment  of  any  curve 
referred  to  rectangular  co-ordinates,  is  equal  to  the  ordinate  into 
the  differential  of  the  abscissa. 

Let  APR  be  a  surface  bounded  by  the  straight  lines  AR, 


Differentials    of   an    Arc,   Area,    etc. 


1R7 


PR,  and  the  arc  AP  of  a  curve  ;  it  is  re- 
quired to  find  the  differential  of  its  area. 

Let  x  represent  the  abscissa  AR,  and 
ij  the  corresponding  ordinate  PR.  If 
we  give  to  x  an  increment  h,  and  make 
RR'=//,  the  value  of  y  will  become  P'R', 
which  we  will  represent  by  y'. 

Since  the  limit  of  the  ratio  of  the  chord  and  arc  of  a  curve 
is  unity,  the  limit  of  the  ratio  of  the  area  included  by  the  ordi- 
nates  PR,  P'R'  and  the  arc  PP',  to  the  trapezoid  included  by 
the  same  ordinates  and  the  chord  PP',  must  be  a  ratio  of 
equality. 

Now  the  trapezoid  PRRT'=RR'xi(PR-i-PR')=i*(y+y')- 
Hence  -, =^.{y+y)- 


h 


But 


Hence 


that  is, 


ax 


cpy  If 


y>=y+-SLh+-^'^-+,  etc.,  Art.  195. 
dz   2 


d\i  h 

PRR'P'_       dyh 

h       ~y+dx  2 


— =!/+-t:7t+»  etc-» 


which  expresses  the  ratio  of  the  increment  of  the  function  to 
that  of  the  variable,  and  we  must  find  the  limit  of  this  ratio  by 
making  the  increment  equal  to  zero,  Art.  174. 

But  in  this  case,  all  the  terms  in  the  second  member  of  the 
equation  which  contain  h  disappear,  and  representing  the  area 
of  the  segment  by  s,  we  have 
ds 


dx 


=y,  or  ds=ydx. 


(254.)  Ex.  To  find  the  differential  of  the  area  of  a  circular 
segment,  take  the  equation 

y-=W-x\ 
Whence  y  =  VIV— x~. 

Hence  ds=ydx=dx  V IV  —  x~. 

The  equation  of  the  circle,  when  the  origin  of  co-ordinates 
is  placed  on  the  circumference,  is 

y=  V2rx—x'\ 
and  hence  the  differential  of  the  area  becomes 
dxV2rx—xi. 


188 


Differential    Calculus. 


Proposition  X, — -Theorem. 

(255.)  The  differential  of  a  surface  of  revolution,  is  equal  to 
the  circumference  of  a  circle  perpendicular  to  the  axis,  multi- 
plied by  the  differential  of  the  arr  of  the  generating  curve. 

Let  the  curve  APP'  be  revolved  about 
the  axis  of  X,  it  will  generate  a  surface 
of  revolution;  and  it  is  required  to  find 
the  differential  of  this  surface. 

Put  AR=a;  and  PR=?/.     If  we  give 
to  x  an  increment  A=RR',  the  value  of  y 
will  become  P'R',  which  we  will  repre 
sent  by  y'. 

In  the  revolution  of  the  curve  APP',  the  points  P  and  P'  will 
describe  the  circumferences  of  two  circles,  and  the  chord  PP' 
will  describe  the  convex  surface  of  a  frustum  of  a  cone.  Also 
since  the  limit  of  the  ratio  of  the  chord  and  arc  of  a  curve  is 
unity,  the  limit  of  the  ratio  of  the  surface  described  by  the 
chord  to  the  surface  described  by  the  arc  must  be  a  ratio  of 
equality. 

Now  the  surface  described  by  the  chord  PP'  is  equal  to 

PP' 

—  X(circ.  PR+«Vc.  P'R'),  Geom.,  Prop.  IV.,  B.  X 

pp 

which  equals  (2-ny-\-2-y'), 


or 


Hence 


PP'XTr(y+3/'). 
the  surface  of  frustum 


PP 


=*(y+y'). 


But 


dy ,     dh/  /i2 
y     yJr~t  +7^9"+'  etc''  Art*  195' 


and 


Hence 


dx 
dlJ-, 


7j'+y=2y+~h+,  etc 
the  surface  of  frustum 


*y* 


pp' 


=7rv%+#+>  etc-), 


which  expresses  the  ratio  of  the  increment  of  the  function  tc 
that  of  the  variable,  and  we  must  find  the  limit  of  this  ratio  by 
making  the  increment  equal  to  zero,  Art.  174. 

But  in  this  case,  all  the  terms  in  the  second  member  of  the 
equation  which  contain  h  disappear,  and  representing  the  arc 


Differentials    of   an    Arc,   Area,   etc. 


18D 


AP  by  z,  and  the  surface  described  by  the  arc  AP  by  S,  we 
have 

— =2ny,  or  dS=2~ydz ; 
dz 

and,  by  substituting  for  dz  its  value,  Art.  251,  we  nave 

d$  =  2mj(dx2+dif)K 

where  2^y  is  the  circumference  of  the  circle  described  by  the 

point  P. 

Proposition  XL — Theorem. 
(256.)   The  differential  of  a  solid  of  revolution  is  equal  to  the 
area  of  a  circle  perpendicular  to  the  axis,  multiplied  by  the  dif- 
ferential of  the  abscissa  of  the  generating  curve. 

Let  the  surface  APR  be  revolved  about 
the  axis  of  X,  it  will  generate  a  solid  of 
revolution,  and  it  is  required  to  find  its 
differential. 

Put  AR=z  and  PR=y.  If  we  give  to 
x  an  increment  A=RR',  the  value  of  y 
will  become  P'R',  which  we  will  represent  by  y'. 

In  the  revolution  of  the  surface  APR',  the  trapezoid  PRR'P 
will  describe  the  frustum  of  a  cone,  and  the  limit  of  its  ratio 
to  the  solid  described  by  the  surface  included  by  the  ordinates 
PR,  PR',  and  the  arc  PP',  is  a  ratio  of  equality. 

Now  the  solidity  of  the  frustum  described  by  the  trapezoid 
PRR'P',  Geom.,  Prop.  VI.,  B.  X.,  is 

j7TxRR'(PR2+PR'2+PRxPR'), 


or 


3' 


h(y*+y"+yy'). 


the  solidity  of  the  frustum     -.,,,„.       A 

ce    _, J-^l — i =\My  +y'  +yy%  . 


Hen 

But 
Hence 

Also, 
Therefore 


du ,     dry  li3 

sf-y+#+^2+'ete-'ArL19B- 


yy'=yt+£ky+> etc- 

ody 


y>+y'*+yy'=3y*+^hy+,  etc.; 


190 


Differential   Calculus. 


that  is, 


solidity  of  frustum     ,  ody 

=M%a+-^Ay+,  etc.), 


h 


which  expresses  the  ratio  of  the  increment  of  the  function  to 
that  of  the  variable,  and  we  must  find  the  limit  of  this  ratio  by 
making  the  increment  equal  to  zero,  Art.  174. 

But  in  this  case,  all  the  terms  in  the  second  member  of  the 
equation  which  contain  h  disappear,  and  representing  the  vol- 
ume of  the  solid  generated  by  V,  we  have 

or  dV=ny*dx, 

where  -ny"1  is  the  area  of  the  circle  described  by  PR. 


DIFFERENTIAL  OF  THE  ARC  AND  AREA  OF  A  POLAR  CURVE 
Proposition  XII. — Theorem. 

(257.)  The  differential  of  an  arc  of  a  polar  curve,  is  equal  to 
the  square  root  of  the  sum  of  the  squares  of  the  differential  of 
the  radius  vector,  and  of  the  product  of  the  radius  vector  by  the 
differential  of  the  measuring  arc. 

Let  PM,  PM'  be  two  radius  vectors  of 
j.  polar  curve,  and  let  MC  be  drawn  from 
iVI  perpendicular  to  PM'.  Then,  in  the 
fight-angled  triangle  M'CM,  we  have 

chord  M'M: 
CM 
CM7 


Also, 
Therefore 


VM'C'+CM2 
tane.  CM'M. 


chord  M'M 


=  Vl+tang.3  CM'M. 


We  must  now  find  the  limit  o:  this  ratio,  by  making  the  in- 
crement of  the  radius  vector  equal  to  zero.  The  limit  of  the 
ratio  of  the  chord  MM'  to  the  arc  MM'  is  unity,  Art.  250. 
Also,  M'C  approaches  to  M'N,  which  is  the  increment  of  the 
radius  vector,  and  the  limit  of  their  ratio  is  unity ;  and  the  an- 

rdt 
gle  CM'M  becomes  PMT,  which  is  equal  to  — ,  Prop.  V.,  Cor 

ar 

Hence,  representing  the  arc  by  z,  we  have 


Asymptotes   of  Curves. 


191 


dz 
dr 


r*dt% 
dr' 


or  dz=Vdrz+ridt\ 

which  is  the  differential  of  the  arc  of  a  polar  curve. 


Pr 


OPOSITION 


XIII.- 


-Theorem. 

(258.)  The  differential  of  the  area  of  a  segment  of  a  polar 
curve,  is  equal  to  the  differential  of  the  measuring  arc,  multiplied 
by  half  the  square  of  the  radius,  vector. 

Let  PMD  be   any  segment  of  a  polar 
curve,  and  let  the  measuring  arc  receive  M^ 
a  small  increment  aa';  the  increment  of 
the  area  will  be  PMM'. 

The  area  of  the  sector  PMN,  Geom., 

Prop.  XII.,  Cor.  B.  VI.,  is  equal  to  MN  X  ^-. 

And  since  aa!  :  MN  : :  1  :  PM, 
MN 
PM" 
sector  PMN     PM2 


aa 


Therefore 

aa'  2 

Now  since  the  limit  of  the  ratio  of  PM'  to  PM  is  a  ratio  of 
equality,  the  limit  of  the  ratio  of  PMM'  to  the  circular  sector 
PMN  is  a  ratio  of  equality.  Taking  the  value  of  this  ratio 
when  the  increment  is  equal  to  zero,  representing  the  segment 
by  s,  and  the  measuring  arc  by  t,  we  have 

ds     r2 

Tt=^ 

.     r*dt 
or  ~2~' 

which  is  the  differential  of  the  area  of  a  segment  of  a  polai 
curve. 


ASYMPTOTES  OF  CURVES. 


(259.)  An  asymptote  of  a  curve  is  a  line  which  continually 
approaches  the  curve,  and  becomes  tangent  to  it  at  an  infinite 
distance  from  the  origin  of  co-ordinates. 

In  some  curves  the  distance  between  the  origin  of  the  co 


*92 


Differential  Calculus. 


ordinates  and  the  point  in  which  the  tangent  meets  the  axes 
increases  continually  with  the  abscissa,  so  that  when  the  ab- 
scissa x  becomes  infinite,  this  distance  is  infinite.  In  other 
curves,  even  when  the  abscissa  becomes  infinite,  the  tangen1 
cuts  the  axes  at  a  finite  distance  from  the  origin.  It  is  then 
called  an  asymptote  to  the  curve. 

Let  A  be  the  origin  of  co-ordinates, 
and  let  TP  be  a  tangent  to  the  curve  at 
a  point  whose  co-ordinates  are  AR=x* 
and  YJi—y.     If  from   the  subtangent 

dx 
TR,  which  equals  y -=-,  the  abscissa  AR 

be  subtracted,  the  remainder, 

AT=y- — x, 
dy 


T 


(1) 


R 


is  the  general  expression  for  AT,  the  distance  from  the  origin 
at  which  the  tangent  intersects  the  axis  of  X. 


Also, 
Hence 


PC=BC  tang.  PBC=%. 
dx 


AB: 


,VK-VC=y-%x, 


(2) 


which  is  a  general  expression  for  AB,  the  distance  from  the 
origin  at  which  the  tangent  intersects  the  axis  of  Y. 

(260.)  If,  when  x  and  y  become  infinite,  either  of  the  ex- 
pressions (1)  and  (2)  reduces  to  a  finite  quantity,  we  may  con- 
clude that  the  curve  has  asymptotes ;  but  if  both  be  infinite, 
then  the  curve  has  no  asymptotes.  If  both  the  expressions  are 
finite,  the  asymptote  will  be  inclined  to  both  the  co-ordinate 
axes  ;  if  one  of  the  values  becomes  finite  and  the  other  infinite 
the  asymptote  will  be  parallel  to  one  of  the  co-ordinate  axes 
if  both  become  zero,  the  asymptote  will  pass  through  the  origin 
of  co-ordinates. 

(261.)  Ex.  1.  It  is  required  to  determine  whether  the  hy- 
perbola has  asymptotes. 

The  equation  of  the  hyperbola,  when  the  origin  of  co-ordi 
nates  is  at  the  center,  is  (Art.  98) 


T>2 


Differentiating,  we  find 


Asymptotes  of  Curves.  19S 

dx  _Ay_x'-A3 
dyV~  Wx~     x     ' 

.  m      dx  Aa 

Therefore  AT=V-j — x= v 

J  ay  x 

Aa 

The  expression represents  the  distance  from  the  origin 

of  co-ordinates  at  which  the  tangent  intersects  the  axis  of  X. 

When  x  is  supposed  infinite,  this  expression  becomes  equai 
to  zero.  Hence  the  hyperbola  has  asymptotes  which  pass 
through  the  center. 

Ex.  2.  It  is  required  to  determine  whether  the  parabola  has 
asymptotes. 

The  equation  of  the  parabola  is 
y*=2px. 

Differentiating,  we  find 

dx    if 
y—=?-=2x. 
dy     p 

dx 
Hence  AT=y- — x=x. 

When  x  is  infinite,  this  quantity  becomes  infinite  ;  therefore  the 
parabola  has  no  asymptotes. 

Ex.  3.  The  equation  of  the  logarithmic  curve  is 
x=\og.  y. 
or  3/=«x- 

Tf  x  be  taken  infinite  and  negative,  then 

1 
y=— =0; 

that  is,  the  axis  of  abscissas  is  an  asymptote  to  the  curve.      See 
fig.,  page  108. 

N 


SECTION  VI. 

RADIUS  OF  CURVATURE— EVOLUTES  OF  CURVES. 

(2G2.)  The  curvature  of  a  curve  is  its  deviation  from  the 
tangent ;  and  of  two  curves  that  which  departs  most  rapidly 
from  its  tangent,  is  said  to  have  the  greatest  curvature. 

Thus,  of  the  two  curves  AC,  AD,  having 
the  common  tangent  AB,  the  latter  de-     ^^-'    "^-^     B 
parts  most  rapidly  from  the  tangent,  and     /  x^0 

is  said  to  have  the  greatest  curvature.  D 

(263.)  The  curvature  of  the  circumference  of  a  circle  is 
evidently  the  same  at  all  of  its  points,  and  also  in  all  circum- 
ferences described  with  equal  radii,  since  the  deviation  from 
the  tangent  is  the  same ;  but  of  two  different  circumferences, 
that  one  curves  the  most  which  has  the  least  radius.  Thus, 
the  circumference  ADF  departs  more 
rapidly  from  the  tangent  line  AB  than 
the  circumference  ACE,  and  this  devia- 
tion increases  as  the  radius  decreases, 
and  conversely.  In  different  circum- 
ferences the  curvature  is  measured  by 
the  angle  formed  by  two  radii  drawn 
through  the  extremities  of  an  arc  of  given  length. 

Proposition  I. — Theorem. 

(264.)  The  curvature  in  two  different  circles  varies  inversely 
as  their  radii. 

Let  R  and  R'  represent  the  radii  of  two  circles,  A  the  length 

of*  a  given  arc  measured  on  the  circumference  of  each ;  a  the 

angle  formed  by  the  two  radii  drawn  through  the  extremities 

of  the  arc  in  the  first  circle,  and  a'  the  angle  formed  by  the 

corresponding  radii  of  the  second.     Then,  by  Geom.,  Prop 

XIV.,  B.  III.,  we  have 

360A 
2ttR  :  A  : :  360°  :  a ;  whence  a—      r>  ; 


Radius   of   Curvature. 


196 


360° 

:  a' ;  whence 

a 

360A 

360A 

a  : : 

2ttR 

'   2ttR" 

a'  :: 

1 
R 

1 
:    R'; 

A 


B 


D 


M 


360A 
and  2-R  :  A  : :  360°  :  a' ;  whence  a' =7—57. 

2ttK 

3f>0  A      SfiO  A 

Therefore 

or 

that  is,  the  curtature  in  two  different  circles  varies  inversely  as 
their  radii. 

(265.)  Let  DBE  be  any  curve  line, 
ABC  a  tangent  at  the  point  B,  and 
B3I  a  normal  at  the  same  point,  then 
will  ABC  be  a  tangent  to  the  circum- 
ference of  every  circle  passing  through 
B,  and  having  its  center  in  the  line 
BM.  The  curve  DBE  may,  therefore, 
be  touched  by  an  infinite  number  of  circles  at  the  same  point 
B.  Some  of  these  circles,  having  a  greater  curvature  than  the 
curve,  fall  wholly  wTithin  it ;  while  others,  having  a  less  degree 
of  curvature,  fall  between  the  curve  and  the  tangent.  Of  this 
infinite  number  of  circles,  there  is  one  which  coincides  most  in- 
timately with  the  curve,  and  is  hence  called  the  oscillatory  cir- 
cle, or  circle  of  curvature,  and  its  radius  is  called  the  radius  of 
curvature  of  the  curve.  The  osculatory  circle  may  be  found 
in  the  following  manner. 

(266.)  Let  there  be  two  curves  wThich 
meet  at  the  point  P,  and  let  us  designate 
the  co-ordinates  of  one  curve  by  x  and 
y,  and  the  co-ordinates  of  the  second 
curve  by  x'  and  y'.  If  we  suppose  x  to 
receive  an  increment  and  become  x+h, 
we  shall  have 


_,_  clii       d~y  If     ffy   h3 

FR'=y  +-fh  +-^  -  +  -jh  —  +,  etc., 
J      ax       dx  2      dx  2.3 


P"R'=y'+^7/i+33i-+33i^+>  etc. 


(1) 
(2) 


dif      dry'  If    d3y'  h5 
dx'       dx"  2     dx'32.3 
But  since  the  point  P  is  common  to  the  two  curves,  we  musi 
have  y—y'' 

Also,  since  the  first  differential  coefficient  represents  the  tan 
ffent  of  the  an^le  which  a  tangent  line  makes  with  the  axis  oi 


"96  Differential  Calculus. 

abscissas,  if  we  suppose  the  two  curves  to  have  a  common 
tangent  at  P,  we  must  have 

ax      ax' 

Now  if  all  the  terms  in  the  first  of  these  developments  are 
equal  to  the  corresponding  terms  in  the  other,  the  curves  will 
be  identical ;  and  the  greater  the  number  of  terms  which  are 
equal  in  the  two  developments,  the  more  intimate  will  be  the 
contact  of  the  curves.  Since  the  general  equation  of  the  circle 
contains  but  three  constants,  the  equality  of  y  and  of  the  first 
and  second  differential  coefficients  in  the  equations  of  the  curve 
and  circle  will  give  three  equations  by  which  the  magnitude 
and  position  of  the  circle  may  be  determined  ;  and  therefore  a 
circle  will  coincide  most  nearly  with  a  given  curve,  when  its  first 
and  second  differential  coefficients  are  equal  to  the  first  and  sec- 
ond differential  coefficients  of  the  equation  of  the  curve. 

(267.)  Since  the  contact  of  the  osculatory  circle  with  a 
curve  is  so  intimate,  its  curvature  is  regarded  as  measured  by 
means  of  the  osculatory  circle.  Thus,  if  we  assume  two  points 
in  the  curve  PP',  and  find  the  radii  r 
and  r'  of  the  circles  which  are  oscu- 
latory at  these  points,  we  shall  have 

curvature  at  P  :  curvature  P' : :  -  :  — ; 

r    r 

that    is,   the    curvature    at   different 

points  varies  inversely  as  the  radius  of  the  osculatory  circle 

Proposition  II. — Theorem. 

(2G8.)  The  radius  of  curvature  at  any  point  of  a  given  curve 
is  equal  to  dz3 

dxdzy 
where  x  and  y  are  the  co-ordinates  of  the  given  point,  and  z 
the  arc  of  the  given  curve. 

The  general  equation  of  the  circle,  Art.  38,  is 

(x-ay+(y-by=n\ 

ivhere  a  and  b  are  the  co-ordinates  of  the  center  of  the  circle 
and  R  is  the  radius. 

Differentiating  this  equation,  and  dividing  by  2,  we  have 

(x  —  a)dx+(y— b)dy=0. 


Radius   of   Curvature.  197 


Differentiating  again,  regarding  dx  as  constant,  we  obtain 
dx'+dy'+^-^dry^O. 

dx'+df 
dy 


ax~-\-ay  .  . 

Whence  y— o= ^ — ,  {}) 


dy(dx*+dy*\ 

and  x-a=Tx\—dY~)'  () 

Substituting  these  values  in  the  equation  of  the  circle,  we 

have 

df(d_£+dyy     (dx*  +  dyy 

R  =dx~\~lhr)  +  \TW~J  ' 


or  R5= 


<Ty     J       \      $y 
(dz'+dy*)' 


whence  R= 


{dxd'yY    ' 
(dx*+dy*Y 


dxd2y 

If  z  denote  the  arc  of  the  given  curve,  then,  Art.  251, 
dz'=dx*+dy' ; 
and  the  above  expression  for  R  becomes 

dxz 

R=-rV",  (3) 

dxdy 
which  is  a  general  expression  for  the  value  of  the  radius  of  the 
osculatory  circle. 

(269.)  To  find  the  radius  of  curvature  for  any  particular 
curve,  we  must  differentiate  the  equation  of  the  curve  twice, 
and  substitute  the  values  of  dx,  dy,  and  dy  in  the  preceding  ex- 
pression for  R.  If  the  radius  of  curvature  for  a  particular  point 
of  the  curve  is  required,  we  must  substitute  for  x  and  y  the  co- 
ordinates of  the  given  point. 

Proposition  III. — Theorem. 

(270.)  The  radius  of  curvature  at  any  point  of  a  conic  sec- 
tion, is  equal  to  the  cube  of  the  normal  divided  by  the  square 
of  half  the  parameter. 

The  general  equation  of  the  conic  sections,  Art.  132,  is 
y^—mx+nx1 ; 

(?n-\-2nx)dx 
whence  dy— — , 

[4f  +  (m+2nxy]dx> 
and  dx*+dy  = ^-j . 


198  Differential   Calculus. 

2nydx1—(m+2nx)dxdy 
Also,       c?y=—2 ^_ > *. 

_[4nj/2— (w+2»x)2]^x2_— m2<fx3 

Substituting  these  values  in  the  equation 

n_(dx-+df)^ 
dxd2y 
we  obtain 


[4(mx+nx-)  +  (7n+2nxy]2 
R=  2nT~  _ ' 

and  dividing  both  terms  of  the  fraction  by  8,  it  becomes 
(  Vmx+nx2+±(m+2nxy) 3 

R= n^ • 


The  numerator  of  this  expression  is  the  cube  of  the  normal, 
Art.  237,  and  the  denominator  is  the  square  of  half  the  pa- 
rameter, Art.  132 ;  that  is,  the  radius  of  curvature  is  equal  to 
the  cube  of  the  normal  divided  by  the  square  of  half  the  pa- 
rameter. 

(271.)  Cor.  1.  The  radii  of  curvature  at  different  points  of 
the  same  conic  section,  are  to  each  other  as  the  cubes  of  the 
corresponding  normals. 

Cor.  2.  If  we  make  x=0,  we  have 

R=— =one  half  the  parameter; 

that  is,  the  radius  of  curvature  at  the  vertex  of  the  major  axis 

of  any  conic  section,  is  equal  to  half  the  parameter  of  that  axis. 

Cor.  3.  If  it  be  required  to  find  the  radius  of  curvature  at 

the  vertex  of  the  minor  axis  of  an  ellipse,  we  make 

2B2  W        .         . 

m=—r-,  n=  — -7-5,  and  x = A, 

A  A 

which  gives,  after  reducing, 

R4; 

that  is,  the  radius  of  curvature  at  the  vertex  of  the  minor  axis 
of  an  ellipse,  is  equal  to  one  half  the  parameter  of  that  axis. 

Cor.  4.  In  the  case  of  the  parabola  in  which  n=0,  the  gen- 
eral value  of  the  radius  of  curvature  becomes 


Evolutes   of  Curves. 


199 


R: 


2??f 


m 


when  x=0,  R=t^  which  is  the  radius  of  curvature  at  the 

vertex  of  the  parabola. 

Ex.  I.  Required  the  length  of  the  radius  of  curvature  for  a 
point  in  a  parabola  whose  abscissa  is  9,  and  ordinate  6. 

Ans. 

Ex.  2.  Required  the  radius  of  curvature  at  the  vertex  of  the 
major  axis  of  an  ellipse  whose  major  axis  is  10  inches,  and 
minor  axis  6  inches.  Ans. 

Ex.  3.  Required  the  radius  of  curvature  at  the  vertex  of  the 
minor  axis  of  the  same  ellipse. 

Ans. 


m  EVOLUTES  OF  CURVES. 

(272.)  An  evolute  is  a  curve  from  which  a  thread  is  sup- 
posed to  be  unwound  or  evolved,  its  extremity  at  the  same  time 
describing  another  curve  called  the  involute. 

Thus,  let  ACC'C"  be  any  curve,  and 
suppose  a  thread,  fastened  to  it  at  some 
point  beyond  C",  is  drawn  tight  to  the 
curve.  Let  it  now  be  gradually  un- 
wound from  the  curve,  keeping  it  al- 
ways tight.  While  the  portion  be- 
tween A  and  C"  is  unwinding,  its  ex- 
tremity will  describe  upon  the  plane 
some  curve  line  APP'P",  the  nature  of 
which  will  depend  on  the  properties 
of  the  other  curve. 

The  curve  ACC'C"  about  which  the 
thread  is  wrapped,  is  called  the  evolute 
of  the  curve  APP"  generated  by  the  extremity  of  the  thread  ; 
and  the  latter  curve  is  called  the  involute  of  the  former. 

(273.)  From  the  manner  in  which  a  curve  is  generated  from 
its  evolute,  we  may  derive  the  following  conclusions: 

1st.  The  portion  of  the  thread  PC,  which  is  disengaged  from 
the  evolute,  is  a  tangent  to  it  at  C. 

2d.  A  tangent  to  the  evolute  curve  at  C  is  perpendicular  to 


200  Differential  Calculus. 

the,  involute  at  the  point  P ;  and  any  point  C  of  the  evolute  may 
be  considered  as  a  momentary  center,  and  the  line  CP  as  the 
radius  of  a  circle  which  the  point  P  is  describing  when  the 
point  of  contact  of  the  tangent  and  curve  is  at  C.  The  points 
C,  C,  C"  are  therefore  the  centers  of  curvature  of  the  points 
P,  P,  P" ;  and  PC,  PC,  P'C"  are  the  radii  of  curvature  of 
the  involute  at  the  points  C,  C,  and  C". 

3d.  The  radius  of  curvature  PC  is  equal  to  the  arc  AC  oj 
the  evolute,  reckoned  from  the  point  A,  where  the  curve  com- 
mences. 

(274.)  Hence,  if  we  suppose  an  osculatory  circle  to  be  drawn 
at  each  of  the  points  of  the  curve  A,  P,  P',  P",  the  centers  of 
all  these  circles  will  be  found  upon  the  curve  ACC'C".  The 
equation  of  the  evolute  is  therefore  the  equation  which  ex- 
presses the  relation  between  the  centers  of  all  the  osculatory 
circles  of  the  involute. 

The  general  equation  of  the  circle,  Art.  38,  is 

(x-ay+(y-by=n\  (i) 

where  a  and  b  denote  the  co-ordinates  of  the  center  of  tn« 
circle. 

To  determine  the  equation  of  the  evolute,  we  must  find  the 
relation  of  a  to  b  in  equation  (1),  regarding  a  and  b  as  the 
co-ordinates  of  the  center  of  the  circle  of  curvature,  and  con- 
sequently the  co-ordinates  of  the  evolute. 

But  we  have  found,  Art.  268,  for  the  circle  of  curvature 

dx*+dy* 

y-b=--^f-,  (2) 

dy 
and  x-a=--^{y-b\  (3) 

and  combining  these  with  the  equation  of  the  involute  curve, 
we  may  obtain  an  equation  from  which  x  and  y  are  eliminated. 

We  must  therefore  differentiate  the  equation  of  the  involute 
twice  ;  deduce  the  values  of  dy  and  d2y,  and  substitute  them  in 
equations  (2)  and  (3) ;  two  new  equations  will  thus  be  obtain- 
ed involving  a,  b,  x,  and  y. 

Combine  these  equations  with  the  equation  of  the  involute, 
and  eliminate  x  and  y ;  the  resulting  equation  will  contain  only 
a  and  b,  and  constants,  and  will  be  the  equation  of  the  solute 
curve. 


Evolutes  of  Curves. 


201 


Ex.  Let  it  be  required  to  find  the  equation  of  the  evolute  of 
the  common  parabola. 
The  equation  of  the  involute  is 

y1=2px  ; 
dy=p 
dx    y' 


whence 


.     p'dx*        ,   ,  p'dz1 

Also,  dyi=J—-r-,  and  <ry=- 


r  "        y 

Substituting  these  values  in  equations  (2)  and  (3),  and  re- 
ducing, we  have 


y 


._y 


y—b=^-+y;  whence  tf= —<t 
*         P  P 


and 


a=  —  ——p. 
P 


Substituting  for  y1  its  value  2px,  we  have 

b2=—;  and  x— a=  —  2x—  p. 
V 
From  this  last  equation  we  derive 

a—  p 


Substituting  this  value  of  x  in  the  preceding  equation,  we 
have 

which  is  the  equation  of  the  evolute,  and  shows  it  to  be  the 
semi-cubical  parabola,  Art.  136. 

If  we  make  6=0,  we  have 
a=p, 
and  hence  the  evolute  meets  the  axis 
of  abscissas,  at  a  distance  AC  from  the 
origin  equal  to  half  the  parameter. 

If  we  transfer  the  origin  of  co-ordi- 
nates from  A  to  C,  the  above  equation 

reduces  to 

8 

27 p 
Since  every  value  of  a  gives  two 
equal  values  of  b  with  contrary  signs,  the  curve  is  symmetrical 


202 


Differential   Calculus, 


with  respect  to  the  axis  of  abscissas.  The  evolute  CM  coi- 
responds  to  the  part  AP  of  the  involute,  and  CM'  to  the  part 
AP'. 


PROPERTIES  OF  THE  CYCLOID. 

(275.)  We  have  found,  Art.  140,  the  equation  of  the  cycloid 
to  be 

x=arc  whose  versed  sine  is  y—  ^2ry—y2. 
The  properties  of  the  cycloid  are  most  easily  deduced  from 
ts  differential  equation,  which  is  obtained  by  differentiating 
Doth  members  of  the  transcendental  equation. 

rdy 


By  Art.  226,  d{arc  whose  versed  sine  is  y) 

rdy-ydy 
V2ry—y2 
rdy-ydy 
V2?-y—y2 


V2ry-y2 


By  Art.  187,  d(-  V2ry-y2)  =  - 


Hence 


dx- 


rdy 


or 


dx= 


V2ry-y"~ 
ydy 


V2ry—y2 
which  is  the  differential  equation  of  the  cycloid. 

(27G.)  If  we  substitute  the  pre- 
ceding value  of  dx  in  the  formulas 
of  Articles  233-6,  we  shall  obtain 
the  values  of  the  tangent  and  subtan- 
gent,  normal  and  subnormal  of  the 
cycloid.     They  are 


subtangent  TR=y~ = 


tangent 


PT: 


subnormal  ~RN=yy-=  V2ry—y", 


normal         PN= 


di/2 
^dx2 


2ry. 


The  subnormal  RN  is  equal  to  PH  of  the  generating  circle 
since  each  is  equal  to  V2ry—y2 ;  hence  the  normal  PN  and 
the  diameter  EN  intersect  the  base  of  the  cycloid  at  the  same 
point. 


Properties   of   tw   Cycloid.  20H 

Proposition  IV. — Theorem. 
(277.)   The  radius  of  curvature  corresponding  to  any  point 
of  the  cycloid,  is  equal  to  double  the  normal. 

If  we  differentiate  again  the  differential  equation  of  the  cy- 

ydy 

cloid,  dz=    — -, 

V2ry— y 

regarding  dx  as  constant,  we  obtain 

i/dy  (rdy — ydy) 

Clearing  of  fractions,  uniting  terms,  and  dividing  by  y,  wo 
have  0={2ry-yi)d*y+rdy*; 

rdy*  rdx* 

whence  ^=-2^=y~~~7~" 

Substituting  the  values  of  dy  and  d?y  in  the  expression  for 
the  radius  of  curvature,  Art.  268, 

(dz2+dyY 


R= 


dxd2y 


(2rdx*\  3 

\    v    J         ail  

we  obtain  R=~~rdx1 — =%2rY=2^2ry- 

~f  _ 

But  we  have  found  the  normal,  Art.  276,  equal  to  V2ry; 

hence  the  radius  of  curvature  is  equal  to  double  the  normal  at 

•.he  point  of  contact. 

Proposition  V. — Theorem. 

(278.)   The  evolute  of  a  cycloid  is  an  equal  cycloid. 

To  obtain  the  equation  of  the  evolute,  we  must  substitute 
the  values  of  dy  and  cFy,  already  found  for  the  cycloid,  in  equa- 
tions (2)  and  (3)  of  Art.  274.     We  thus  obtain 

(2ry-y*)dx2 
dx*+- 


t    h_    dx*+(W= t 

J  d'y  rdx* 


y' 

_/+2ry-y> 


204 

Whence 
Also, 

Whence 


Differential   Calculus. 


y=-h. 


x-a=~T^y-hy 


\f2ry—y' 


X2y, 


=  —  2V2ry—y* 


x=a—2V2ry—y*. 
Substituting  these  values  of  x  and  y  in  the  transcendenla' 
equation  of  the  cycloid,  Art.  140, 

x=arc(ve?,sed  si?ie=y)~  V2ry—y*, 
we  obtain 

a— 2  V— 2rb— b'= arc  (versed  sine——b)—  V  —  2rb—b*, 
or  a=arc(versed  sine=—b)  +  V  —  2rb— b'\ 

which  is  the  transcendental  equation  of  the  evolute  referred  to 
the  primitive  origin  and  the  primitive  axes. 

This  is  also  the  equation  of  a 
cycloid  whose  generating  cir- 
cle is  equal  to  that  of  the  given 
one,  and  whose  vertex  coincides 
witn  the  extremity  of  the  base, 
lying,  however,  below  the  base, 
as  appears  by  substituting  —  b 
for  y  in  the  equation  of  Art.  140. 

Thus,  the  evolute  AA'  of  the  cycloid  is  an  equal  cycloid  ; 
the  arc  AA'  is  identical  with  AB,  and  the  vertex  B  is  trans- 
ferred to  A. 


SECTION   VII. 

ANALYSIS  OF  CURVE  LINES. 

(279.)  If  it  was  possible  to  resolve  an  equation  of  any  de- 
gree, we  might  fol.ow  the  course  of  a  curve  represented  by 
any  Algebraic  equation,  by  methods  explained  in  Analytical 
Geometry.  By  assigning  to  the  independent  variable  different 
values,  both  positive  and  negative,  we  could  determine  any 
number  of  points  of  the  curve  at  pleasure. 

The  Differential  Calculus  enables  us  to  abridge  this  investi- 
gation, and  may  be  employed  even  when  the  equation  of  the 
curve  is  of  so  high  a  degree  that  we  are  unable  to  obtain  a  gen- 
eral expression  for  one  of  the  variables  in  terms  of  the  other. 

The  first  object  aimed  at  in  such  an  analysis,  is  to  discover 
those  points  of  a  curve  which  present  some  peculiarity;  such 
as  the  point  at  which  the  tangent  is  parallel  or  perpendicular 
to  the  axis  of  abscissas.  Such  points  have  been  named  singu- 
lar points..  A  singular  point  of  a  curve  is  one  which  is  dis- 
tinguished by  some  remarkable  property  not  enjoyed  by  the 
other  points  of  the  curve  immediately  adjacent. 

Proposition  I. — Theorem. 

(280.)  For  a  point  at  which  the  tangent  to  a  curve  is  paral- 
lel to  the  axis  of  abscissas,  the  first  differential  coefficient  is  equal 
to  zero. 

For  the  first  differential  coefficient  expresses  the  value  of 
the  tangent  of  the  angle  which  the  tangent  line  forms  with  the 
axis  of  abscissas,  Art.  201  ;  and  when  this  line  is  parallel  with 
the  axis,  the  angle  which  it  forms  with  the  axis  is  zero,  and  its 
tangent  is  zero. 

Proposition  II. — Theorem. 

(281.)  For  a  point  at  which  the  tangent  to  a  curve  is  perpen- 
di'-ular  to  the  axis  of  abscissas,  the  first  differential  coefficient  is 
equal  to  infinity. 

For  the  first  differential  coefficient  expresses  the  vaiue  of 


206  Differential   Calculus. 

the  tangent  of  the  angle  which  the  tangent  line  forms  with  the 
axis  of  abscissas ;  and  when  this  angle  is  90  degrees,  its  tan- 
gent is  infinite. 

Ex.  1.  Itts  required  to  determine  at  what  point  the  tangent 
to  the  circumference  of  a  circle  is  parallel  to  the  axis,  and 
where  it  is  perpendicular. 
Take  the  equation 

x*+y*=R\ 
By  differentiating,  we  obtain 

dy        x 
dx        y' 
and  placing  this  equal  to  zero,  we  find 

x=0. 
But  when  x=0,  we  have 

y=±R; 

hence  the  tangent  is  parallel  to  the  axis  of  abscissas  at  the  two 

points  where  the  circumference  intersects  the  axis  of  ordinates. 

If  we  make 

dy        x  y     „ 

■/== — =oo,  or  --=0, 
dx        y  x 

we  find  y=0.     But  when  y—0,  we  have 

x=±R; 

that  is,  the  tangent  is  perpendicular  to  the  axis  of  abscissas  at 
the  two  points  where  the  circumference  intersects  the  axis  of 
abscissas. 

Ex.  2.  It  is  required  to  determine  at  what  point  the  tangent 
to  a  cycloid  is  parallel  to  the  base,  and  when  it  is  perpendicu- 
lar to  the  base. 

Proposition  III. — Theorem. 

(282.)  If  a  curve  is  convex  toward  the  axis  of  abscissas,  the 
ordinate  and  second  differential  coefficient  will  have  the  same 
sign. 

Let  PP'P"  be  a  curve  convex  toward  the  axis  of  abscissas ; 
and  let  x  and  y  be  the  co-ordinates  of  the  point  P.  Let  x  be 
increased  by  any  arbitrary  increment  RR',  which  we  will  rep- 


Analysis   of   Curve   Lines. 


207 


resent  by  k,  and  take  R'R"  also  equal 
to  h.  Draw  the  ordinates  P'R',  P"R" ; 
draw  the  line  PP\  and  produce  it  to 
B;  join  P'  and  P",  and  draw  PD, 
P'D'  parallel  to  AR.  We  shall  then 
have 

PR=*/, 


p,R,=y+ £+£-+,  etc. 

Also, 

dy  ,     cT-yAh2 


Hence 


dx        dx* 
P'D=P'R'-PR= 


(1) 


(2) 


R 


n' 


R" 


dv ,     dry  h*  ,„N 


Subti acting  equation  (1)  from  equation  (2),  we  have 

P'D'=P"R"-P,R'=^A+^^-+,  etc.  (4) 
dx      dx    2 

Subtracting  equation  (3)  from  equation  (4),  remembering 
that  BD'  is  equal  to  P'D,  we  have 

P'B=P"D'-P'D=^/r+,  etc.  (5) 

Now  when  we  suppose  h  to  be  taken  indefinitely  small,  the 
sign  of  the  second  member  of  equation  (5)  will  depend  upon 
that  of  the 'first  term;  and  since  the  first  member  of  the  equa- 
tion is  positive,  the  second  must  also  be  positive ;  that  is,  the 
second  differential  coefficient  is  positive  ;  and  the  ordinate,  be- 
ing situated  above  the  axis  of  abscissas,  is  also  positive. 

If  the  curve  is  below  the  axis  of  abscissas,  we  shall  have 

dry 
-p"b=p"d'-p'd=-7+h'i+,  etc. ; 

and  since  the  first  member  of  this  equation  is  negative,  the  sec- 
ond will  also  be  negative ;  that  is,  the  second  differential  coef- 
ficient is  negative.  Whence  we  conclude  that  if  the  curve  is 
convex  toward  the  axis  of  abscissas,  the  second  differential  co- 
efficient will  be  positive  when  the  ordinate  is  positive,  and  neg- 
ative when  the  ordinate  is  negative. 


208 


Differential  Calculus. 


Proposition  IV. — Theorem. 

(283.)  If  a  curve  is  concave  toward  the  axis  of  abscissas,  the 
ordinate  and  second  differential  coefficient  will  have  contrary 
signs. 

Let  PP'P"  be  a  curve  concave  to- 
ward the  axis  of  abscissas ;  and  let 
x  and  y  be  the  co-ordinates  of  the 
point  P.  Let  x  be  increased  by  any 
arbitrary  increment  RR',  which  we 
will  represent  by  h,  and  take  R'R"  A 
also  equal  to  h.  Draw  the  ordinates 
P'R\  P"R"  ;  draw  the  line  PP',  and 
produce  it  to  B.  Join  P'  and  P", 
and  draw  PD,  P'D'  parallel  to  AR. 
We  shall  then  have 

?R=y, 


B 

P>^ 

D' 

*?y 

D 

/ 

R 

R' 

R" 

\ 

d 

?\ 

V 

d' 

V 

^ 

X 

b 

„  „  dii       d2y  Aa 


Also, 


P^=y+|2A+^+,etc. 


d*y4W 
2 


0) 
(2) 
(3) 


Hence      P'D=P'R'-PR=^A+^  |+,  etc. 

Subtracting  equation  (1)  from  equation  (2),  we  have 

P»D'=P"R"-P'R'=A+^  ^+,  etc.   (4) 
dx      dx    2 

Subtracting  equation  (3)  from  equation  (4),  remembering 

that  BD'  is  equal  to  P'D,  we  have 


d2v 
P"B=P"D'-P,D=-p^+>  etc. 


(5) 


Now  since  the  first  member  of  this  equation  is  negative,  the 
second  member  must  also  be  negative ;  that  is,  the  second  dif- 
ferential coefficient  will  be  negative,  while  the  ordinate  is  posi- 
tive. 

If  the  curve  is  below  the  axis  of  abscissas,  we  shall  have 

d*V 

+jt,''Z,=p"^_J0^=^/i2+,  etc., 

where  the  second  differential  coefficient  is  positive,  while  the 
ordinate  is  negative. 


Analysis   of  Curve   Lines.  209 

Ex.  1.  It  is  required  to  determine  whether  the  circumference 
of  a  circle  is  convex  or  concave  toward  the  axis  of  abscissas 
The  equation  of  the  circle  is 

x2+y3=Ra; 

dy        x 

whence  -7-— — 

ax        y 

$y_    x->+if_     R» 

A1S0'  dx>~        y*     ~     y"     ■ 

which  is  negative  when  y  is  positive,  and  positive  when  y  is 
negative.  Hence  the  circumference  is  concave  toward  the 
axis  of  abscissas. 

Ex.  2.  It  is  required  to  determine  whether  the  circumference 
of  an  ellipse  is  convex  or  concave  toward  the  axis  of  abscissas. 

(284.)  Definition.  A  point  of  inflection  is  a  point  at  which 
a  curve  from  being  convex  toward  the  axis  of  abscissas,  be- 
comes concave,  or  the  reverse. 

Proposition  V. — Theorem. 

For  a  point  of  inflection,  the  second  differential  coefficient 
must  be  equal  to  zero  or  infinity. 

When  the  curve  is  convex  toward  the  axis  of  abscissas,  the 
ordinate  and  second  differential  coefficient  have  the  same  sign, 
but  when  the  curve  is  concave,  they  have  contrary  signs. 
Hence,  at  a  point  of  inflection,  the  second  differential  coeffi- 
cient must  change  its  sign.  Therefore,  between  the  positive 
and  negative  values  there  rousl  be  one  value  equal  to  zero  or 
infinity  ;  and  the  roots  of  the  equation 
dry  d2y 

d?^0rdx-^' 
will  give  the  abscissas  of  the  points  of  inflection. 

Having  discovered  that  the  second  differential  coefficient  for  a 
certain  point  of  a  curve  is  equal  to  zero  or  infinity,  we  increase 
and  diminish  successively  by  a  small  quantity  h,  the  abscissa 
of  this  point;  and  if  the  second  differential  coefficient  has  con- 
trary signs  for  these  new  values  of  x,  we  conclude  that  here 
is  a  point  of  inflection. 

Ex.  1.  Determine  whether  the  curve  whose  equation  is 
y=a  +  (x— b)3, 
has  a  point  of  inflection. 

0 


210 


Differential   Calculus. 


By  differentiating,  we  find 


and 


£■*-* 

g-*-* 


A 


When  x=b,  the  first  differential  coefficient  is  zero,  and  the 
tangent  is  parallel  to  the  axis  of  abscissas  at  the  point  whose 
co-ordinates  are  x=b,  y=a. 

When  x<b,  the  second  differential  coefficient  is  negative; 
but  when  x>b,  the  second  differential  coefficient  is  positive; 
that  is,  the  second  differential  coefficient  changes  its  sign  at 
the  point  of  the  curve  of  which  the  ab- 
scissa is  x=b;  consequently  there  is  an  in- 
flection of  the  curve  when  x=b. 

On  the  left  of  P  the  curve  falls  below 
the  tangent  line  TT',  while  on  the  right  of 
P  it  runs  above  TT'. 

Ex.  2.  Determine  whether  the  curve  whose  equation  is 
y=a—  (x— b)3, 
has  a  point  of  inflection. 

Ans.  The  curve  is  first  convex  and 
then  concave  toward  the  axis  of  ab- 
scissas, and  there  is  an  inflection  at  the 
point  x=b. 

Ex.  3.  Determine  whether  the  curve  whose  equation  is 
y=3x+18x*-2x% 
has  a  point  of  inflection. 
By  differentiating,  we  find 

-l=3+3Gx-6x\ 
ax 


—  T' 


and 


^=36-12z. 
ax2 


Putting  the  second  differential  coefficient  equal  to  zero,  we 
obtain  x=3. 

Take,  therefore,  AB=3,  and 
draw  the  ordinate  BC :  C  is  the 
point  of  inflection.  If  x  be  be- 
tween 0  and  3,  36—  12a;  is  posi- 


Analysis   of   Curve   Lines.  211 

tive ;  therefore  the  part  AC  of  the  curve  is  convex  to  AB ; 
but  when  x  is  greater  than  3,  36—  12x  is  negative,  and  there- 
fore the  curve  is  concave  toward  the  axis. 

(285.)  Definition.  A  multiple  point  is  a  point  at  which  two 
or  more  branches  of  a  curve  intersect  each  other. 

Proposition  VI. — Theorem. 

For  a  multiple  point,  the  first  differential  coefficient  must  have 
several  values. 

It  is  obvious  that  where  two  branches  of  a  curve  intersect, 
there  must  be  two  tangents  which  have  different  values  ;  and 
since  the  first  differential  coefficient  expresses  the  tangent  of 
the  angle  which  the  tangent  makes  with  the  axis  of  abscissas, 
this  coefficient  must  have  as  many  values  as  there  are  inter- 
secting branches. 

For  a  multiple  point,  the  first  differential  coefficient  generally 

reduces  to  the  form  of  -,  which  represents  an  indeterminate 
quantity,  Algebra,  Art.  130. 

Ex.  1.  It  is  required  to  determine  whether  the  curve  repre- 
sented by  the  equation 

y*=a*x9—x*, 

has  a  multiple  point. 

Extracting  the  root  of  each  member,  we  have 

y=±x{a*-xy.  (1) 

By  differentiating,  we  obtain 

dy      ,    a*-2x* 

•  dx=± 1'  <2> 

ax         (a'-xY 

We  see  from  equation  (1)  that  every  value  of  x  gives  two 
values  of  y  with  contrary  signs ;  hence  the  curve  has  two 
branches,  which  are  symmetrical  with  respect  to  the  axis  of  X. 
Also,  when  x=±a,  y=Q ;  that  is,  the  curve  cuts  the  axis  of 
x  at  the  points  B  and  C,  at  the  distances  +a  and  —a  from  the 
origin.  When  x=0,  y=0 ;  hence  the  twf>  branches  intersect 
at  the  origin  A,  which  is  therefore  a  multiple  point.  At  this 
point  there  are  two  tangents  given  by  equation  (2),  which, 
when  x=0,  reduces  to 


212 


DlFFERENTIAS     CALCULUS. 


dx 


=±a. 


Hence  one  tangent  line  makes 
an  angle  with  the  axis  of  abscissas 
whose  tangent  is  +a,  the  other  an 
angle  whose  tangent  is  —a. 

Ex.  2.  It  is  required  to  determine  whether  the  curve  repre« 
cented  by  the  equation 

y-^(x-a)\x-b), 

has  a  multiple  point. 

Ans.  The  point  whose  co-ordinates  are  x=a,  y=0,  is  a  mul- 
tiple point. 

(286.)  Definition'.  A  cusp  is  a  point  at  which  two  or  more 
branches  of  a  curve  terminate  and  have  a  common  tangent. 
If  the  branches  lie  on  different  sides  of  the  tangent,  it  is  called 
a  cusp  of  the  first  order ;  if  both  branches  lie  on  the  same  side 
of  the  tangent,  it  is  called  a  cusp  of  the  second  order. 

Since  the  axes  of  reference  may  be  chosen  at  pleasure,  we 
shall,  for  convenience,  suppose  the  tangent  at  a  cusp  to  be  per- 
pendicular to  the  axis  of  abscissas.  If  the  tangent  is  parallel 
to  the  axis  of  abscissas,  we  have  but  to  transpose  the  terms 
abscissa  and  ordinate  in  the  two  following  theorems. 


Proposition  VII. — Theorem. 

(287.)  A  point  of  a  curve  at  which  the  tangent  is  perpendicu- 
lar to  the  axis  of  abscissas,  and  the  contiguous  ordinates  on 
each  side  of  that  point  are  real,  and  both  greater  or  both  less 
than  the  ordinate  of  the  given  point,  is  a  cusp  of  the  first  order 

If  P  be  a  point  of  a  curve  at  which  the 
tangent  is  perpendicular  to  AX,  and  if  the 
ordinates  P'R',  P'R",  however  near  they 
may  be  taken  to  PR,  are  both  greater  than 
PR,  it  is  evident  that  P  will  be  the  point 
of  meeting  of  two  branches  which  have 
PR  for  their  common  tangent,  as  repre- 
sented in  the  annexed  figure. 

If  P'R',  P'R"  are  both  less  than  PR,  P  will  be  the  point  of 
meeting  of  two  branches  which  have  PR  for  their  common 


A 


R'llll" 


■X 


Analysis   of  Curve   Lines. 


213 


tangent,  but  the  branches  will  be  situated 
as  in  the  figure  annexed. 

Ex.  1.  It  is  required  to  determine  wheth- 
er the  curve  represented  by  the  equation 

y=a+2(x-by, 

has  a  cusp  of  the  first  order. 
By  differentiating,  we  obtain 

dy  4 


R'RR" 


dx  3(x-by 

When  x=b,  this  coefficient  becomes  infinite,  and  the  tangent 
will  be  perpendicular  to  the  axis  of  abscissas  at  the  point  whose 
co-ordinates  are  x=b,  y=a. 

Let  us  now  substitute  for  x,  in  the  equation  of  the  curve, 
b+h  and  b—h  successively;  we  shall  obtain  in  each  case 

a 

y=a+2h3 ; 

and  hence  y  is  less  when  x=b,  than  for  the  adjacent  values  of 
x  either  greater  or  less  than  b.  Hence  there  is  a  cusp  at  the 
point  whose  co-ordinates  are  £=&,  y=a. 

Ex.  2.  It  is  required  to  determine  whether  the  curve  repre- 
sented by  the  equation 

y=a-2(x-bf, 

has  a  cusp  of  the  first  order. 

If  we  substitute  for  x,  in  the  equation  of  the  curve,  b+h  and 
b—h  successively,  we  shall  obtain  in  each  case 

2 

y=a— 2h3, 

and  hence  y  is  greater  when  x=b,  than  for  the  adjacent  values 
of  x  either  greater  or  less  than  b.  Hence  there  is  a  cusp  at 
the  point  whose  co-ordinates  are  x=b,  y=a. 

Ex.  3.  It  is  required  to  determine  whether  the  curve  repre- 
sented by  the  equation 

x*=y\ 

has  a  cusp  of  the  first  order. 


214 


Differential  Calculus. 


Proposition  VIII. — Theorem. 

(288.)  A  point  of  a  curve  at  which  the  tangent  is  perpendicu- 
la?'  to  the  axis  of  abscissas,  and  the  contiguous  abscissa  upon 
one  side  of  the  given  point,  has  two  values  both  greater  or  both 
less  than  the  abscissa  of  the  given  point,  is  a  cusp  of  the  second 
order. 

If  P  be  a  point  of  a  curve  at  which 
the  tangent  is  perpendicular  to  AX,  and  & 
if  corresponding  to  the  ordinate  AR', 
there  are  two  abscissas  P'R',  P"R',  both 
greater  than  PR,  however  near  they 
may  be  taken  to  PR,  it  is  evident  that 
P  is  the.  point  of  meeting  of  two  branches 
which  have  PT  for  their  common  tan- 
gent, as  represented  in  the  annexed  figure. 

If  P'R',  P"R'  are  both  less  than  PR, 
P  will  be  the  point  of  meeting  of  two  E.' 
branches  which  have  PT  for  their  com- 
mon tangent,  but  the  branches    will  be 
situated  as  in  the  figure  annexed. 

Ex.  It  is  required  to  determine  wheth- 
er the  curve  represented  by  the  equation 
i 

has  a  cusp  of  the  second  order. 
By  differentiating,  we  obtain 

dy_       1 

iy±W~ 

We  see  from  the  equation  of  the  curve  that  the  curve  has 
two  branches,  both  of  which  pass  through  the  origin  of  co-or- 
dinates. When  y=0,  x—0,  and  the  first  differential  coefficient 
reduces  to  infinity  ;  and  hence  the  axis  of  Y 
ordinates  is  tangent  to  both  branches  of  the 
curve  at  the  origin  of  co-ordinates.  If  y  is 
supposed  to  be  negative,  x  is  imaginary; 
hence  the  curve  does  not  extend  below  the 
axis  of  abscissas. 

If  we  suppose  y—+h,  we  shall  have 


R 


Analysis  of  Curve  Lines.  215 

x=h*±h\ 

5 

When  k  is  less  than  unity,  h2  is  less  than  h\  and  x  will  have 
two  positive  values,  PR  and  P'R  ;  hence  the  point  A  is  a  cusp 
of  the  second  order. 

By  a  similar  course  of  investigation,  the  cusps  may  be  de- 
termined when  the  tangent  is  inclined  to  both  the  co-ordinate 
axes. 

(289.)  Definition.  An  isolated  point  is  a  point  whose  co-or- 
dinates satisfy  the  equation  of  a  curve,  while  the  point  itself  is 
entirely  detached  from  every  other  in  the  curve. 

Proposition  IX. — Theorem. 

For  an  isolated  point,  the  first  differential  coefficient  is  equal 
to  an  imaginary  constant. 

For  since,  by  supposition,  the  proposed  point  is  entirely  de- 
tached from  every  other  point  of  the  curve,  there  can  be  no 
tangent  line  corresponding  to  that  point,  and  consequently  the 
value  of  the  first  differential  coefficient  must  be  imaginary. 

Ex.  It  is  required  to  determine  whether  the  curve  represent- 
ed by  the  equation 

y'i=x(a+x)2, 
has  an  isolated  point. 

Extracting  the  square  root,  we  find 

yz=±(a+x)  y/X. 

Hence,  when  x  is  negative,  y  will  be  imaginary.  If  x=0,  y=0, 
which  shows  that  the  curve  passes  through  the  origin  A.  For 
every  positive  value  of  x,  y  will  have  two  real  values,  which 
shows  that  the  curve  has  two  branches  extending  indefinitely 
toward  the  right. 

The  equation  is  also  satisfied  by  the 
values  £=  —  a,  y—0.  Hence  the  point  P, 
whose  abscissa  is  —a,  is  detached  from 
all  others  in  the  curve,  and  is  called  an 
isolated  point.  The  form  of  the  curve  is 
*uch  as  exhibited  in  the  annexed  figure. 

(290.)  From  the  preceding  propositions  it  will  be  seen  that, 
in  order  to  trace  out  a  curve  from  its  equation,  we  first  dis- 


216         Differential  Calculus. 

cover  the  most  remarkable  points  by  putting  x  and  y  success- 
ively equal  to  zero  or  infinity,  and  also  the  first  and  second 
differential  coefficients  equal  to  zero  or  infinity.  Then,  to  trace 
the  curve  in  the  neighborhood  of  the  points  thus  determined, 
when  they  appear  to  present  any  peculiarity,  we  increase  one 
of  the  co-ordinates  by  a  very  small  quantity,  and  observe  the 
effect  upon  the  other  co-ordinate.  Having  determined  the 
singular  points,  and  examined  the  course  of  the  curve  in  their 
'mmediate  vicinity,  we  can  easily  trace  the  remainder  of  the 
curve,  by  assigning  to  x  and  y  arbitrary  values  at  pleasure. 


INTEGRAL  CALCULUS. 


SECTION    I. 

INTEGRATION  OF  MONOMIAL  DIFFERENTIALS  —  OF  BINOMIAL 
DIFFERENTIALS  —  OF  THE  DIFFERENTIALS  OF  CIRCULAR 
ARCS. 

Article  (291.)  The  Integral  Calculus  is  the  reverse  of  the.* 
Differential  Calculus,  its  object  being  to  determine  the  expres- 
sion or  function  from  which  a  given  differential  has  been  de- 
rived. 

Thus  we  have  found  that  the  differential  of  x*  is  2xdx 
therefore,  if  we  have  given  2xdx,  we  know  that  it  must  have 
been  derived  from  x\  or  x*  plus  a  constant  term. 

(292.)  The  function  from  which  the  given  differential  has 
been  derived,  is  called  its  integral  Hence,  as  we  are  not  cer- 
tain whether  the  integral  has  a  constant  quantity  or  not  added 
to  it,  we  annex  a  constant  quantity  represented  by  C,  the  value 
of  which  is  to  be  determined  from  the  nature  of  the  problem. 

(293.)  Leibnitz  considered  the  differentials  of  functions  as 
indefinitely  small  differences,  and  the  sum  of  these  indefinitely 
small  differences  he  regarded  as  making  up  the  function  ;  hence 
the  letter  S  was  placed  before  the  differential  to  show  that  the 
sum  was  to  be  taken.  As  it  was  frequently  required  to  place 
S  before  a  compound  expression,  it  was  elongated  into  the  sign 
/,  which,  being  placed  before  a  differential,  denotes  that  its  in- 
tegral is  to  be  taken.     Thus, 

f2xdx=x2+C. 

This  sign  /  is  still  retained  evei  i  by  those  who  reject  the 
philosophy  of  Leibnitz. 

(294.)  We  have  seen  that  the  differential  coefficient  ex- 
presses the  ratio  of  the  rate  of  variation  of  the  function  to  that 
of  the  independent  variable.  Hence,  when  we  have  given  a 
certain  differential  to  find  its  integral,  it  is  to  be  understood 


218  Integral   Caiculus. 

that  we  have  given  a  certain  quantity  which  varies  uniformly, 
and  the  ratio  of  its  rate  of  variation  to  another  quantity  de- 
pending on  it  and  given  quantities,  to  find  the  value  of  that 
quantity. 

Thus,  if  we  have  given 

du=3x2dx, 
to  find  its  integral,  we  have  given  a  quantity  x  which  varies 
uniformly,  and  the  ratio  of  its  rate  of  variation  to  that  of  u,  to 
find  the  value  of  u.     And  since 

f3x*dx=x*, 
we  know  that  u  equals  x3,  or  x3+C 

Ex.  There  is  a  quantity  x  which  increases  uniformly,  and 
the  rate  of  its  variation,  compared  with  another  quantity  de- 
pending on  it,  is  as  1  to  ax" ;  required  the  value  of  this  quan- 
tity when  a=9  and  £=10. 

Let  u—the  quantity  required. 

Then  dx  :  du  : :  1  :  ax*. 

Hence  du=ax'2dx, 

and  fdu=fax*dx, 

ax9 

u=—. 

Hence  the  number  required  is 

fXl03=3000. 

(295.)  We  have  seen  (Art.  176)  that  the  differential  of  the 
product  of  a  variable  multiplied  by  a  constant,  is  equal  to  the 
constant  multiplied  by  the  differential  of  the  variable.  Hence 
we  conclude  that  the  integral  of  any  differential  multiplied  by 
a  constant  quantity,  is  equal  to  the  constant  multiplied  by  the 
integral  of  the  differential. 

Thus,  since  the  differential  of  ax  is  adx,  it  follows  that 
fadx =  ax= afdx. 
Hence, 

Proposition  I. — Theorem. 

If  the  expression  to  be  integrated  have  a  constant  factor,  this 
factor  may  be  placed  without  the  sign  of  integration. 

Thus,  fabx'2dx=abfxidx. 

(296.)  We  have  seen  (Art.  179)  that  the  differential  of  a 
function  composed  of  several  terms  is  equal  to  the  sum  or  dif 


Integration   of  Monomial  Differentials.    219 

ferenoe  of  the  differentials  taken  separately.  Hence  the  in- 
tegral of  a  differential  expression  composed  of  several  terms  is 
equal  to  the  sum  or  difference  of  the  integrals  taken  separate- 
ly.    Thus,  since  the  differential  of 

a*x*—  2ax3— x 
is  2a2xdx—6ax2dx—dx, 

we  conclude  that     f(2a2xdx  —  6ax'dx—dx) 
is  aixi—2ax3—x. 

Hence  we  derive 

Proposition  II. — Theorem. 

The  integral  of  the  sum  or  difference  of  any  number  of  diffe? 
entials,  is  equal  to  the  sum  or  difference  of  their  respective  in- 
tegrals. 

(297.)  We  have  seen  (Art.  177)  that  every  constant  quan- 
tity connected  with  the  variable  by  the  sign  plus  or  minus 
will  disappear  in  differentiation  ;  that  is,  the  differential  of  u  -J-C 
is  the  same  as  that  of  u.  Consequently,  the  same  differential 
may  answer  to  several  integral  functions,  differing  from  each 
other  only  in  the  value  of  the  constant  term.     Hence 

Proposition  III. — Theorem. 

In  integrating,  a  constant  term  must  always  be  added  to  the 
integral. 

Thus,  fdu=u+C. 

(298.)  We  have  found  (Art.  186)  that  the  differential  of 

xm+1  is  (m+l)xmdx. 

dxm+1       (  zm+1  \ 
Hence  xmdx= — rr=d[  — rr  )  • 

?n+l        \m-\-\J 

xm+1 

Therefore is  the  function  whose  differential  is  xmdx,  or 

m-\-\ 

xm+1 
fxmdx = — — - + C. 
J  m+\ 

Hence 

Proposition  IV. — Theorem. 

To  find  the  integral  of  a  monomial  differential  of  the  form 
xmdx,  increase  the  exponent  of  the  variable  by  unity,  and  then 
divide  by  the  new  exponent  and  by  the  differential  of  the  variable. 

Ex.  1.  The  rate  of  variation  of  the  independent  variable  .r, 


220  Integral   Calculus. 

is  to  the  rate  of  variation  of  a  certain  algebraic  expression  aa 

I  to  j-x9 ;  it  is  required  to  find  that  expression. 


x9dx 
Ex.  2.  What  is  the  integral  of  — —  ? 


Ex.  3.  What  is  the  integral  ofx2dx? 


A  aX  r» 

Ans.  —  +C. 


x 
Ans.  — -fC. 
y 


Ans.  §:c3-|-C. 


dx  i 

Ex.  4.  What  is  the  integral  of  —  or  x  2dx  ? 

°  ^x 


Ans.  2x2+C. 


dx 
Ex.  5.  What  is  the  integral  of  —  or  x~3dx  ? 


x~2  1 

Ans.  — —  or  —  r-^+C. 
2  2x 

dv 
Ex.  6.  What  is  the  integral  of  ax*dx+——%  ? 

ax3       '     . , 
Ans.  — — {-x'+L. 

O 

Ex.  7.  If  the  side  of  a  square  increases  uniformly  at  the  rate 
ot  T\  of  an  inch  per  second,  what  is  the  area  of  the  square 
when  it  is  increasing  at  the  rate  of  a  square  inch  per  second  ? 

Ans. 

(299.)  There  is  one  case  in  which  the  preceding  rule  fails. 
It  is  that  in  which  the  exponent  m  is  equal  to  —1.  For  in  this 
case  we  have,  according  to  the  rule, 


fx  ldx- 


x 


-1+1       x°     1 


=— =r=ao. 


-1+1      0     0 

which  shows  that  the  rule  is  inapplicable. 

dx 
But  x  1dx  is  the  same  as  — ,  and  we  know  (Art.  215)  that 

x  v 

this  expression  was  obtained  by  differentiating  the  logarithm 

of  the  denominator.     Therefore 


fdx 
fx~ldx  or  /  — =log.  .r-t-C. 


Integration  of  Monomial  Differentials.   221 

.  .                                /•  adx 
Also,  / —a  log.  x+C 

Hence  we  have 

Proposition  V. — Theorem. 

If  the  numerator  of  a  fraction  is  the  product  of  a  constant 
*  quantity  by  the  differential  of  the  denominator,  its  integral  is  the 
product  of  the  constant  by  the  Naperian  logarithm  of  the  de- 
nominator. 

dx 
Ex.  1.  What  is  the  integral  of  — — ? 

°  a+x 

Ans.  log.  (a+x)-\-G. 

%bxdx 
Ex,  2.  What  is  the  integral  of  — -r-=  ? 
&  a+bx2 

Ans.  log.  (a-\-bx2)-\-C. 

adx 
Ex.  3.  What  is  the  integral  of  -j—  ? 

Ans. 

ax^dx 
Ex.  4.  What  is  the  integral  of — =—•? 

°  x3 

Ans. 

Proposition  VI. — Theorem. 
(300.)  Every  polynomial  of  the  form 

(a-\-bx-\- cx2+,  etc.)ndx, 
\n  which  n  is  a  positive  whole  number,  may  be  integrated  by 
raising  the  quantity  within  the  parenthesis  to  the  nth  power, 
multiplying  each  term  by  dx,  and  then  integrating  each  term 
separately. 

This  is  an  obvious  consequence  of  Proposition  II. 

Ex.  1.  What  is  the  integral  of  (a+bx)~dx  ? 

Expanding  the  quantity  within  the  parenthesis,  and  multi- 

■  plying  each  term  by  dx,  we  have 

a'dx + 2abxdx + b2x2dx. 

Integrating  each  term  separately,  we  obtain 

&V 
a'x+abx2+-^-+C, 

o 

which  is  the  integral  sought. 


222  Integral   Calculus. 

Ex.  2.  What  is  the  integral  of  (5+7a?)*dx? 

Ans. 
Ex.  3.  What  is  the  integral  of  (a+3xydx? 

Ans. 
(301.)  We  have  seen  (Art.  188)  that  any  power  of  a  poly- 
nomial may  be  differentiated  by  diminishing  the  exponent  of 
the  power  by  unity,  and  then  multiplying  by  the  primitive  ex- 
ponent and  by  the  differential  of  the  polynomial.  Thus  the 
differential  of  (ax+x*)3  is 

3(ax+x'iy(adx+2xdx). 
Hence  we  deduce 

Proposition  VII. — Theorem. 

In  order  to  integrate  a  compound  expression  consisting  of  any 
power  of  a  polynomial  multiplied  by  its  differential,  increase  the 
exponent  of  the  polynomial  by  unity,  and  then  divide  by  the  new 
exponent,  and  by  the  differential  of  the  polynomial. 

Ex.  1.  What  is  the  integral  of  (a+3xyGxdx  ? 

Ans.  <2+5£T+c. 

o 

Ex.  2.  What  is  the  integral  of  (2x3-l)6x'dx  ? 

Ans. 

(302.)  The  preceding  rule  is  equally  applicable  when  the 
exponent  of  the  polynomial  is  fractional. 

Ex.  3.  What  is  the  integral  of  (x+ax)2(dx+adx)  ? 

3 

Ans.  %(x+ax)2+C. 

dx 
Ex.  4.  What  is  the  integral  of jl 

{1+xf 

Ans. 
Ex.  5.  What  is  the  integral  of  (ax*+bxsy(2ax+3bx2)dx? 

Ans. 
i 
Ex.  6.  What  is  the  integral  of  {ax+bx'1f{a-\-2bx)dx? 

Ans. 
(303.)  Any  binomial  differential  of  the  form 
du = ( a + bxn)  mxn~ldx, 
in  which  the  exponent  of  the  variable  without  the  parenthesis 


Integration   of  Binomial   Differentials.    223 

is  one  less  than  the  exponent  of  the  variable  within,  may  be 
integrated  in  the  same  manner. 

Let  us  put  y=a+bxn. 

Then  dy=bnx*~ldx, 

1  d^         n-lJ 

-and  t~=x     dx. 

on 

_,        .  ,  dy 

Therefore  du=ym1-, 

*  bn 

and  u=-. — — rr-. — |-C, 


(a+bxT+1  , 
Hence  we  deduce 


(m+l)bn 

{a+bx*)m't 
{in  +  \)bn 


Proposition  VIII.—Theorem. 

To  integrate  a  binomial  differential  when  the  exponent  of  the 
variable  without  the  parenthesis  is  one  less  than  that  within,  in- 
crease the  exponent  of  the  binomial  by  unity,  and  divide  by  the 
product  of  the  new  exponent,  the  coefficient,  and  the  exponent  of 
the  variable  within  the  parenthesis. 

Ex.  1.  What  is  the  integral  of  du=(a+3x*yxdx  ? 
Let  us  put  y=a+3x* ; 

dy 
whence  dy=6xdx,  or  xdx=—. 

b 

y3dy 
Therefore  du=^—^-, 

D 

y*     (a+Sx-y 
and  tt=_=___+C. 

i 

Ex.  2.  What  is  the  integral  of  (a-\-bxf)2mxdx  ? 


m  i 

Ans.  -Aa+bxY+C. 


xdx 
Ex.  3.  What  is  the  integral  of 


vV+*a 


Ans.  Va'+x'+C. 

3 

Ex.  4    Whit  is  the  integral  of  (a+bxy~exdx? 

56 


224  Integral   Calculus. 

Ex.  5.  What  is  the  integral  of  (cf+xyanx^dx  ? 

Ans. 
Ex.  6.  If  x  increase  uniformly  at  the  rate  of  one  inch  pei 
second,  what  is  the  form  and  value  of  the  expression  which  is 

\-\-x 
ncreasing  at  the  rate  of  -==  inches  per  second,  when 
°  V2x+x* 

x=  10  inches?  Ans. 

(304.)  To  complete  each  integral  as  determined  by  the  pre- 
ceding rules,  we  have  added  a  constant  quantity  C.  While 
the  value  of  this  constant  is  unknown,  the  expression  is  called 
an  indefinite  integral.  But  in  the  application  of  the  calculus  to 
the  solution  of  real  problems,  the  complete  value  of  the  integral 
is  determined  by  the  conditions  of  the  problem.  We  may  de- 
termine the  value  of  the  constant,  or  make  it  disappear  entire- 
ly from  the  integral,  in  the  following  manner.  If  we  suppose 
the  independent  variable  and  the  integral  to  begin  to  exist  at 
the  same  instant,  then  when  x=0,  the  integral  =0,  and  conse- 
quently C=0. 

Again,  if  we  suppose  the  integral  to  begin  to  exist,  or  to 
nave  its  origin  when  x  becomes  equal  to  a  given  quantity  a, 
the  value  of  C  may  then  be  determined. 

When  the  value  of  the  constant  has  been  determined,  and  a 
particular  value  assigned  to  the  independent  variable,  the 
value  of  the  integral  is  then  known,  and  is  called  a  definite  in- 
tegral. 

Ex.  1.  Represent  the  base  of  the  tri-  r 

angle  ABC  by  x,  and  the  perpendicular  „. 

by  nx,  then  the  area  of  the  triangle  is  E/^i 

\nx*,  whose  differential  is  >*f      \ 

nxdx.  /^        \        \ 

If  we  take  the  integral  of  nxdx  accord-  A  D      F       B 

ing  to  Prop.  IV.,  we  obtain 

fnxdx = hix* + C, 
which  represents  the  area  of  the  triangle  ABC. 

The  constant  is  determined  by  observing  that  the  base  x, 
and  the  area  of  the  triangle  begin  to  exist  at  the  same  time ; 
hence  when  x=0,  the  integral  =0  ;  that  is, 

\nx2+G  =  0, 
and  consequently  C=0. 


Integration    of   Binomial   Differentials.  22a 

Again,  suppose  we  wish  to  obtain  an  expression  for  the  area 
of  the  trapezoid  EDFG,  contained  between  the  two  perpendic- 
ulars DE,  FG.  We  must  first  obtain  the  area  of  the  triangle 
ADE.  Suppose  the  variable  x  to  be  equal  to  AD,  which  we 
^ill  represent  by  a,  then  the  area  of  ADE  will  be  expressed  by 

Next  suppose  the  variable  x  to  become  equal  to  AF,  which  we 
will  represent  by  b,  then  the  area  of  AFG  will  be  expressed  by 

±nb*+C. 

Subtracting  the  former  expression  from  the  latter,  we  obtain 
the  area  of  the  trapezoid  EDFG, 

\nb^— \na*. 

(305.)  Hence  we  find  that  the  constant  C  may  be  made  to  dis- 
appear by  giving  two  successive  values  to  the  independent  varia- 
ble, and  taking  the  difference  between  the  two  mtegrals  corre 
sponding  to  these  values. 

When  we  take  the  excess  of  the  value  of  an  integral  when 
the  independent  variable  has  become  equal  to  b,  above  its  value 
when  it  was  only  equal  to  a,  we  are  said  to  integrate  between 
the  limits  of  x—a  and  x=b. 

This  is  indicated  bv  the  sign  /    . 

«/  a 


l^,^        !  „_,*(*-«') 


Thus,  /    nxdx=\nb2—\na'i= 

Ex.  2.  Integrate  /    2xdx,  and  illustrate  the  case  by  a  geo> 

metrical  example. 

Ans. 

Ex.  3.  Integrate  /    3x-dx ;  illustrate  the  case  by  a  geomet 

rical  example,  and  determine  its  numerical  value  when  a=4 
and  6=6. 

Ans. 

Ex  4.  Integrate  /  -xdx ;  illustrate  the  case  by  a  geomet- 
rical example,  and  determine  the  value  of  the  definite  integral 
between  the  limits  a=2  and  6=3. 

-4.71  s. 

Ex.  5.  Integrate  /     -x'dx ;  illustrate  the  case  by  a  geomet- 


22Q  Integral   Calculus. 

rical  example,  and  determine  the  value  of  the  definite  integral 
between  the  limits  a=4  and  6=6. 

Ans. 

Ex.  6.  What  is  the  value  of  /   2(e+x)dx,  when  a=W,  b=20. 

and  e=4  ?  and  illustrate  the  exercise  by  a  geometrical  figure. 

Ans. 

Ex.  7.  What  is  the  value  of  /    ^{e+nx^^nxdx,  when  a=4, 

b=6,  e=4,  and  n=2l 

Ans. 

pb  dx 
Ex.  8.  What  is  the  value  of  /     — - — ,  when  a— 2,  b=3,  and 

J  a  e+x 

e=4l 

Ans. 

INTEGRATION  BY  SERIES. 
(306.)  If  it  is  required  to  integrate  an  expression  of  the  form 

in  which  X  is  a  function  of  a:,  it  is  often  best  to  develop  X  into 
a  series,  and  then,  after  multiplying  by  dx,  to  integrate  each 
term  separately.  This  is  called  integrating  by  series,  since  we 
thus  obtain  a  series  equal  to  the  integral  of  the  given  expres- 
sion, from  which  we  may  deduce  the  approximate  value  of 
the  integral  when  the  series  is  a  converging  one. 

dx 

Ex.  1.  It  is  required  to  integrate  the  expression  t^t-. 

By  the  binomial  theorem,  we  find 

or  (l+x)~1  =  l—x+xi—x3-\-x*—,  etc. 

l+x 

Multiplying  by  dx,  we  have 

=dx—xdx-\-x'idx—xsdx+xidx—,  etc. ; 

l+x 

and  integrating  each  term  separately,  we  obtain 


A 


if^-l+l-7+ir-' etc- +c- 


dx 
Ex.  2.  It  is  required  to  integrate  the  expression  „ 


Differentials  of  Circular   Arcs  227 

By  the  binomial  theorem,  we  find 

or  (l+x*)-1=l-x*+x*-x'i+,  etc. 


1+x- 


dx 

Whence        =dx— x"dx+x*dx  — xadx+,  etc., 

l+a;2 

/'  dx  x3    xb    x1  ,  _ 

1-+p-*-8  +  677+»etC''+a 

dx 

Ex.  3.  What  is  the  integral  of ? 

°  a—x 

Ans. 

dx 

Ex.  4.  What  is  the  integral  of ^  ? 

°        .    (a— a;) 

JLws. 

dx 


Ex.  5.  What  is  the  integral  of 


Vl-x* 

Ans. 


INTEGRATION  OF  THE  DIFFERENTIALS-  OF  CIRCULAR  ARCS. 

(307.)  We  have  found  in  Art.  227,  that  if  z  designates  an 
arc,  and  y  its  sine,  the  radius  of  the  circle  being  unity, 

dz=-^=. 
Vl-if 

Hence  f  r  V     =z+C. 

If  the  arc  be  estimated  from  the  beginning  of  the  first  quad- 
rant, the  sine  will  be  0  when  the  arc  is  0,  and  consequently  C 
equals  zero. 

Therefore  the  entire  integral  is 

/I     =the  arc  of  which  y  is  the  sine. 
Vl-y* 

If  it  were  required  to  integrate  an  expression  of  the  form 

Va  -y 
it  may  be  done  by  the  aid  of  an  auxiliary  variable. 

y 

Assume  v=-  or  y=av. 


Then  dy=adv^aud  Vd2—y2—aVl  —  v^ 


S28 


Integral  Calculus. 


Substituting  these  values  in  equation  (1),  we  have 

,  dv 

dz=- 


Hence 
or 

Ex.  Integrate  the  expression 
dz 


Vl-v- 

z=the  arc  whose  sine  is  v, 


y 

2= the  arc  whose  sine  is  -. 
a 


dy 


V4-f 

Ans.  z=the  arc  whose  sine  is  \y. 
(308.)  We  have  found  in  Art.  227,  that  if  z  designates  an  arc. 
and  y'  its  cosine,  the  radius  of  the  circle  being  unity, 

-dy' 


dz- 


Hence 


Vl-y" 
J  Vl-v'2 


To  determine  the  constant  C,  we  see  that 
if  the  arc  be  estimated  from  B,  the  beginning 
of  the  first  quadrant,  the  cosine  becomes  0 
when  the  arc  becomes  a  quadrant,  which  is 
represented  by  \it  ;  hence  the  first  member 
of  the  equation  becomes  equal  to  \tt  when 
y'  =  0.  But  under  this  supposition  the  arc 
whose  cosine  is  0  becomes  \tt\  hence  C  =  0,  and  the  entire  in 
tegral  is 

-dy' 


f- 


:the  arc  whose  cosine  is  y'. 


Vl-y" 
it  it  were  required  to  integrate  an  expression  of  the  form 

Vcf-y'2 
it  may  be  done  by  the  aid  of  an  auxiliary  variable,  as  in  Art. 
307,  and  we  shall  find 

•     •  y' 

z=the  arc  whose  cosine  is  — . 

a 

(309.)  We  have  found  in  Art.  227,  that  if  z  designates  an 

arc,  and  t  its  tangent, 

dt 

dz~TTf' 


Differentials   of  Circular  Arcs.  229 

Hence  /lT?=Z+a 

If  the  arc  is  estimated  from  the  beginning  of  the  first  quad- 
rant,  we  shall  have 

%=0  when  I  rXT^0  '■>  hence  C=0, 
and  the  entire  integral  is 

— — =the  arc  of  which  t  is  the  tangent. 
If  it  were  required  to  integrate  an  expression  of  the  form 

a  +t 

it  may  be  done  by  the  aid  of  an  auxiliary  variable. 

t 
Assume  v=- or  t=av; 

then  dt=adv. 

Substituting  in  equation  (1),  we  have 

adv        1/  dv   \ 
^=a2  +  aV~a\l+uV 

Hence  z=-X arc  whose  tangent  is  v, 

a 

1  .    t 

or  z=-Xarc  whose  tangent  is  -. 

a  a 

(310.)  We  have  found,  Art.  227,  that  if  %  designates  an  arc, 

and  x  its  versed  sine, 

dx 
ax—  , — ==■ 

V2x—x* 

/'     dx  n 

V2x^=Z+C- 
If  the  arc  is  estimated  from  the  beginning  of  the  first  quad« 
rant,  we  shall  have 

C  =  0, 
and  the  entire  integral  is 

/'     dx 
=the  arc  of  which  x  is  tte  versed  sine. 
y/2x— x2 

If  it  were  required  to  integrate  an  expression  of  the  form 


230  Integral  Calculus. 

*=-7==.  CO 

V2ax— x 

it  may  be  done  by  the  aid  of  an  auxiliary  variable. 

Assume  v=—  or  x=av  ; 

a 

then  dx=adv. 

Substituting  in  equation  (1),  we  have 
adv  dv 


dz=- 


V2a*v-a*v*     V2v-v* 
Hence  z=the  arc  whose  versed  sine  is  v, 

x 

or  z=the  arc  whose  versed  sine  is  -. 

a 


INTEGRATION  OF  BINOMIAL  DIFFERENTIALS. 

Proposition  IX. — Theorem. 

(311.)  Every  binomial  differential  can  be  reduced  to  the  form 

p 
xm-l{a-\-bxnydx, 

in  which  the  exponents  m  and  n  are  whole  numbers,  and  n  is 
positive. 

1st.  For  if  m  and  n  were  fractional,  and  the  binomial  were 
of  the  form 

i  \  p 

x3(al-\-bx2)'1dxi 

we  may  substitute  for  x  another  variable,  with  an  exponent 
equal  to  the  least  common  multiple  of  the  denominators  of  the 
given  exponents,  by  which  means  the  proposed  binomial  will 
be  transformed  into  one  in  which  the  exponents  of  the  variable 
are  whole  numbers. 

Thus,  if  we  make  x=z*,  we  find 

x^(a+bxfydx=6z\a+bzydz, 
in  which  the  exponents  of  z  are  whole  numbers. 

2d.  If  n  were  negative,  or  the  expression  were  of  the  form 

p 
xm~\a-itbx-ydx 

we  may  put  x=-,  in  which  case  we  shall  obtain 


Integration   of   Binomial   Differentials.  231 

-z-m-\a+bzydz, 
in  which  the  exponent  of  the  variable  within  the  parenthesis 
is  positive. 

3d.  If  the  variable  x  were  found  in  both  terms  of  the  bino- 
mial, and  the  expression  were  of  the  form 

p 
xm-l(axT+bx*ydz, 

we  may  divide  the  binomial  within  the  parenthesis  by  x\  and 

EI 

multiply  the  factor  without  the  parenthesis  by  x*,  and  we  shall 
obtain 

xm+T-\a+bxn-ydx, 
in  which  but  one  of  the  terms  within  the  parenthesis  contains 
the  variable  x. 

Proposition  X. — Theorem. 

(312.)  Every  binomial  differential,  in  which  the  exponent  of 
the  parenthesis  is  a  whole  number  and  positive,  can  be  integrated 
by  raising  the  quantity  within  the  parenthesis  to  the  proposed 
power,  multiplying  each  term  by  the  factor  without  the  paren 
thesis,  and  then  integrating  each  term  separately. 

This  results  directly  from  Proposition  II. 
Ex.  1.  Integrate  the  expression 

du=x*(aJrbx*ydx. 
Expanding  the  binomial,  we  obtain 

du=dix'idx + 2abxf'dx + bqx*dx. 
A.nd  integrating  each  term  separately,  we  find 

aV     abx°     6V     _ 

u= 1 1 r-C. 

3         3         9 

Ex.  2.  Integrate  the  expression 

du=x*(a+bx*)3dx. 

aV    3a*bx*    SaVx*    b'x" 
Ans.  ^=^+-^-+-g-+lo'+C 

Ex.  3.  Integrate  the  expression 

du=x*(a+bxydx. 
Ans. 


232  Integral  Calculus. 

Ex.  4.  Integrate  the  expression 

du=xb(a+b1x*)sdx. 
Ans. 

Proposition  XL — Theorem. 

(313.)  Every  binomial  differential  can  be  integrated,  when  the 
exponent  of  the  variable  without  the  parenthesis,  increased  by 
unity,  is  exactly  divisible  by  the  exponent  of  the  variable  within. 

For  this  purpose,  we  substitute  for  the  binomial  within  the 
parenthesis,  a  new  variable  having  an  exponent  equal  to  the 
denominator  of  the  exponent  of  the  parenthesis. 

Let  us  assume  a+bxn=z\ 

Then  (a+bzn)*=z*.  (1) 

2q — a 
Also,  xn= — - — 

and 

and,  by  differentiating, 


-PrO* 


mx 


^-a^c^r* 


(2) 


Multiplying  together  equations  (1)  and  (2),  and  dividing  by 
m,  we  obtain 

m 

xm-1(a+bxydx=-jz^-1{—r-j      dz, 
which,  according  to  Prop.  X.,  can  be  integrated»when  —  is  a 


whole  number  and  positive.     If  —  is  negative,  we  may,  by 

Formula  D,  Art.  323,  increase  the  exponent  until  it  becomes 
positive. 
Ex.  1.  Integrate  the  expression 

3 

du=x3(a  +  bx1)  2dx. 
Assume  a+bx*=z'i. 

Then  (a+bxy=z\  (1) 

Also,  x%=—jT  (2) 


Integration   of  Binomial  Differentials.  233 

7      zdz  /o\ 

and  xdx=-j-.  \<*) 

Multiplying  together  equations  (1),  (2),  and  (3),  we  obtain 
du=x\a+bz*)sdx=z*  .—^-dz. 

zT     azb 
Hence  u=W~5b2 

Replacing  the  value  of  z,  we  find 

(a+bxrf    a(a+bx*y  ,  n 
lb  56 

Ex.  2.  Integrate  the  expression 

du=x\a+bx*)2dx. 
Assume  a+bx2=z\ 

Then  (a+bx*)*=z.  (1) 

Also,  X^Z~T'  •  (2) 

7       zdz  /o\ 

and  xdx—-r--  {#) 

/z*-a\'z*dz 
Hence  du—  1 — -, —  J  -r-, 

z0(Zz  —  2az  4dz +a'z'^g 
or  ^w = fi  "* 

z7     2«z5    aV 
and  M=W-5F+W+U 

Restoring  the  value  of  z,  we  find 


(a+bz*  7 


(a+ZQ3     2a(a+&:ca)2     a*(a+bx*)2 
W  563  363 


Ex.  3.  Integrate  the  expression 

du=x*(a+bxvfdx. 


_s\5 


3(a+bx>)  *      3a(a+bx*)3     3a\a+bx')'s  ,  n 
Ans.  u=——y g^        +        ^         .  U 

Ex.  4.  Integrate  the  expression 

..I 
du=x*(a—x'2)  2dx. 

If  we  put  a— x*=z\  we  find 


234  Integral   Calculus. 

du=  —  (a  —  z*)dz. 

z* 
Whence  u=—  az+— +C, 

o 

or  u——a{a—x)-\ hC 

o 

Ex.  5.  Integrate  the  expression 

du=x\a?+x*)-'dx. 
If  we  put  z=aa+;cQ,  we  find 

zdz  a4dz 

and  m=- — «"z+—  log.  2+C-, 

4  2 


or 


/      9    I        S\    9  4 

U=(12±)  -a'la'+x^+^log.W+x^+C. 

Proposition  XII. — Theorem. 

(314.)  Every'  binomial  differential  can  be  integrated,  when 
the  exponent  of  the  variable  without  the  parenthesis,  augmented 
by  unity,  and  divided  by  the  exponent  of  the  variable  within  the 
parenthesis,  plus  the  exponent  of  the  parenthesis,  is  a  whole 
number. 

p 
The  binomial  xm~1(a+bxn)']dx,  may  be  written 

x^-'U^+bJx^^dx, 

p    np 

or  xm-\ax-n  +  bfx*dx, 

m[np  x  p 

which  equals  x     q     (ax~" +b)qdx, 

which,  according  to  the  preceding  Proposition,  can  be  inte- 


grated when 


np 
m+-L 


—  is  a  whole  number, 


or  (  — r-—  J  is  a  whole  number. 

\n     qj 

Ex.  1    Integrate  the  expression 

du=a(\-\-x*)    2dx. 


Integration   of  Binomial  Differentials.  235 


Put  vV=l+z'. 

Then  (l+x2)-1^-9^"3-  <*> 

1 


Also, 


Whence  xtf  —  lY' 

and  1=*>'-1)'.       .  (3) 

Multiplying  together  equations  (1),  (2),  and  (3),  we  have 

Jw=a(l+^2)  2dx  =  — jr. 
Whence  a=-=-==+C 

Ex.  2.  Integrate  the  expression 

_j 

<Ztt=:zr4(1  —  ^)  2dx' 

put  »V==1— a;9. 

Then  aTt=»"+ll 

and  x-^=(v'+\y.  (1) 

Also,  ^(u'  +  l)""8. 

_3 

Whence  <fo=-(u2+l)  2^f.  (2) 

Also,                    (1-*')  2=-=S—  (3) 

Multiplying  together  equations  (1),  (2),  and  (3),  we  have 
du=x-*(l-a?)~^dx=-(v*+l)dv. 
Whence  a=_— _i>= — +u, 

r+2x2  - — j,r 

or  «= £T"  . 

Ex.  3.  Integrate  the  expression 

_i  dx 

put  »=a;+Va!+x!. 


Then  i^dx+-^=^  . ■Vrf+tf 

Therefore  V=7^+?' 


2S6  Integral   Calculus. 

Consequently  we  have 

X0=  / —  =  /  — =log.  v=\og.  [x+  VoM^1]. 

J    Va'+x2    J    v 

Ex.  4.  Integrate  the  expression 

*dx 


c?X2=- 


Va2+x2 


Put  D=(aV+^)2. 

__,                                   _      a2xdx+2xsdx 
VV  hence  dv= — , 

(aV+.r4)2 

a?dx  2x*dx 

or  dv= f+- 


(a'+z2)*     (a2 +:£*)* 
or  e?u=a'</X0+2^X3, 

where  X0  has  the  same  value  as  in  Ex.  3. 

wn                                ,rv      dv     a*dX" 
Whence  dX2=— 


2  2 

B     a2X„ 
and  JL=- — 


2        2   ' 

or  X2=-(a2+:c2)2--X0. 

(315.)  When  a  binomial  differential  can  not  be  integrated  by 
either  of  the  preceding  methods,  its  integral  may  be  made  to 
depend  upon  the  integral  of  another  differential  of  a  simpler 
form.  This  is  effected  by  resolving  the  binomial  into  two 
parts,  one  of  which  has  a  known  integral. 
We  have  seen,  Art.  180,  that 

d(uv)  =  udv  +  vdu  ; 
whence,  by  integrating, 

uv=fudv  +fvdu, 
and  consequently, 

fudv=uv—fvdu,  (1) 

a  formula  which  reduces  the  integration  of  udv  to  that  of  vdu, 
and  which  is  known  by  the  name  of  integration  by  parts. 

For  greater  convenience,  we  shall  represent  the  binomial 
differential  by  xm(a+bxn)vdx,  where  p  is  supposed  to  be  a  frac- 
tion, but  m  and  n  are  whole  numbers. 


Integration    of   Binomial   Differ iintials.  237 

Proposition  XIII. — Theorem. 
(316.)   The  integral  of  any  differential  of  the  form 
xm(a-\-bxn)pdx, 
may  be  made  to  depend  upon  the  integral  of  another  differential 
of  the  same  form,  but  in  which  the  exponent  of  the  variable  with- 
out the  parenthesis  is  diminished  by  the  exponent  of  the  variable 
within  the  parenthesis. 

Let  us  put  v=(a+bxn)% 

where  s  is  an  exponent  to  which  any  value  may  be  assigned, 

as  may  be  found  most  convenient. 

Differentiating,  we  find 

dv = bnsxn~l  {a + bxn) s_  War. 

If  then  we  assume 

udv=xm (a+bxn)pdx,  Art.  315, 

xm-T,+1(a+bxy-s+1 

we  must  have  u= : , 

bns 

and,  by  differentiating, 

,       (?n-n+l)xm-D(a+bxy-s+1  ,       (p-s+l)xm(a+bxy , 

dit = ; dx  ^ dx 

bns  s 

But  (a+bxny-s+1=(a+bxn)(a+bxny-a 

=  a(a+bxny-s+bx\a+bxny-f. 

Hence 

Ya(m— n-\-l)xm~a     (??i+l+np  —  ns)xml,       ,     .     ,7 

du=    — r +- - —    (a+bxy-'Jx. 

L  bns  ns  J 

Let  the  value  of  s  be  taken  such  that 

m-\-\+np— ns=0 ; 

and  we  shall  have 


m+1 
that  is,  s= \-p, 


,      a(m—n+l)a?a-n(a+bx"Y~*, 

du = — —  dx. 

b(jip+m+l) 

Substituting  the  values  of  u,  v,  du,  and  dv,  here  given,  in 
formula  (1),  Art.  315,  we  obtain 

Formula  A. 

,     ,      ,     v    ,      xm-D+l(a+bxny+1-a(?n-n+l)fxm-D(a-{-bxnydx 

fxm(a+bxnydx= * '- ;/      ,       ,  ,/•  ■ — 

'      v  '  b(np+m+l) 


238  Integral   Calculus. 

\ 

by  which  the  integral  of  the  proposed  differential  is  made  to 
depend  upon  the  integral  of 

in  which  the  exponent  of  the  factor  xm  without  the  parenthesis, 
is  diminished  by  that  of  xn,  the  variable  within  the  parenthesis ; 
and  by  a  similar  process  we  should  find 
fx^ia  +  bxydx, 
to  depend  on  fxm~'in{a-]rbxaydx ; 

and  by  continuing  this  process,  the  exponent  of  the  factor  with- 
out the  parenthesis  may  be  diminished  until  it  is  less  than  n. 

(317.)  We  have  frequent  occasion  to  integrate  binomial  dif- 
ferentials of  the  form 

xmdx 
dXm=   ,  u 

vV-z2 

This  may  be  done  by  Formula  A,  Art.  316,  by  substituting 

aa  for  a, 

-1  for  b, 

2  for  n, 

-iforp. 

Whence  we  obtain 

Formula  a. 
=  f   x-dx        (m-l)a'  Cx^dx  _x^_  ^-^ 
m    J  Va'-x*  m      J  vV-z2       m 

Ex.  1.  Integrate  the  expression 


Va'  —  x* 


We  have  found  in  Art.  226,  that     tinoV   ,  is  the  differential 

of  an  arc  of  a  circle  of  which  R  is  the  radius  and  yis  the  sine. 
Hence  dX0  is  the  differential  of  a  circular  arc,  and  X0  is  the 
arc  of  a  circle  of  which  a  is  the  radius  and  x  is  the  sine. 

Ex.  2.  Integrate  the  expression 

x2dx 
dX.2= 


vV-z2 
Make  m,  in  Formula  a,  equal  to  2,  and  the  formula  reduces  to 


¥ 


Integration    of    Binomial   Differentials.  289 


X2=-X--vV-:r2, 

where  X0  has  the  same  value  as  in  Ex.  1. 
Ex.  3.  Integrate  the  expression 


vV-.r2 

Make  m,  in  Formula  a,  equal  to  4,  and  the  formula  reduces  to 
3a2         x3 


X4=— X2-—  vV-rc2, 

where  X2  has  the  same  value  as  in  Ex.  2. 
Ex.  4.  Integrate  the  expression 


Vcf-x* 
Make  m,  in  Formula  a,  equal  to  6,  and  the  formula  reduces  to 

X6=— X4-  — vV-a;2, 

where  X4  has  the  same  value  as  in  Ex.  3. 
Ex.  5.  Integrate  the  expression 

a. 


vV-z2 

Make  m,  in  Formula  a,  equal  to  8,  and  the  formula  reduces  to 
7a2 


X8=— X6— —  \/a2— a;2, 

where  X6  has  the  same  value  as  in  Ex.  4. 

(318.)  Formula  a  reduces  the  binomial  differential 

xmdx 


!-. 


Va*-x* 
to  that  of 


r  xm~2dx  _ 
J  vV^' 


and,  in  a  similar  manner,  this  is  found  to  depend  upon 

xm~*dx 


f 


Va*-x* 


and  so  on ;  so  that  after  —  operations,  when  m  is  an  even  num 
ber,  the  integral  is  found  to  depend  upon 


240  Integral   Calculus,. 

dx 


f 


Va2-x* 

x 


whicn  represents  a  circular  arc  whose  sine  is  -,  Art.  307. 
r  a 

In  a  similar  manner  is  derived 

Formula  b. 

Ex.  Integrate  the  expression 


Va'+x1 

Make  m=4,  in  Formula  &,  and  it  reduces  to 


~X.l=—Va2+x 


3a2  r   x*d 


4  4./    VaJ+xa 

The  integral  of  =  has  been  given  in  Art.  314,  Ex.  4. 

(319.)  The  binomial 


V2ax— x* 

may  be  integrated  by  means  of  Formula  A,  Art.  316,  by  making 
the  proper  substitutions.  It  may,  however,  be  integrated  by 
an  independent  process  as  follows  : 


Let  us  put  v=xm~l  V'2ax-x%  or  (2ax2m_,-2rm)2 
Differentiating,  we  find 

a{2m-  l)x'm-2dx-?nx"°-1dx 


dv- 


(2ax'm-1-x,2m)2 
a{2m—\)xm-*dx        mxmdx 


(2ax-xY  (2ax-xy 

But  this  last  term  is  equal  to  m.dXm  ;  hence 
a(2m—l)xm-idx    dv 


dXm=- 

7il 


m(2ax—x*)2 
and,  by  integrating  we  obtain 


Integration    of   Binomial   Differentials.  24 1 

Formula  c. 

/'     ^  a<2"-1'  f    X""dx    -*—  V2^=? 

J   y/2ax-xL  m       J   V2ax—x"      ™ 

vvlr-ch   diminishes   the  exponent  of  the  variable  without  the 
parenthesis  by  unity. 

Ex.  1.  Integrate  the  expression 


V2ax—x" 


Rdx  ,.„ 

We  have  found  in  Art.  226,  that— ======  is  the  difterentia! 

■J'ZRx—x' 

of  an  arc  of  a  circle  of  which  R  is  the  radius  and  x  is  the  versed 

sine.     Hence  dX0  is  the  differential  of  a  circular  arc,  and  X0  is- 

the  arc  of  a  circle  of  which  a  is  the  radius  and  x  is  the  versed 


sine. 


Ex.  2.  Integrate  the  expression 
dX,= 


V2ax  -f 

Make  m,  in  Formula  c,  equal  to  unity,  and  the  formula  re 
duces  to 

X1=X0—  V2ax—x'2, 
where  Xa  has  the  same  value  as  in  Ex.  1. 
Ex.  3.  Integrate  the  expression 

dX,- 


V2ax—x2 
Make  ??i,  in  Formula  c,  equal  to  2,  and  the  formula  reduces  to 


3a' 


Xi=-Xl--V2ax~x\ 

where  Xi  has  the  same  value  as  in  Ex.  2. 
Ex.  4.  Integrate  the  expression 

dX>= 


V2ax—x2 
Make  ?n,  in  Formula  c,  equal  to  3,  and  the  formula  reduces  to 


5a, 


X3=—  X-^-V2ax-  x> 
3  o 

wnere  X,  has  the  same  value  as  in  Ex.  3. 

Q 


242  Integral   Calculus. 

Ex.  5.  Integrate  the  expression 


V2ax—xi 
Make  m,  in  Formula  c,  equal  to  4,  and  the  formula  reduces  to 


la. 


X4=-— X3 — -V2ax—x\ 
4  4 

where  X3  has  the  same  value  as  in  Ex.  4. 

(320.)  Formula  c  reduces  the  differential  binomial 

xmdx 


I 


y/2ax—xt 

f   xm~1dx 

to  that  of  /  -; 

J   V2ax  —  x2 

and,  in  a  similar  manner,  this  will  be  found  to  depend  upon 

xm~*dx 


S: 


I a'  ' 

v  2ax— x 

and  so  on  ;  so  that  after  in  operations,  when  m  is  a  whole  num- 
ber, the  integral  is  found  to  depend  upon 

dx 


f 


y/2ax—xi 

.    x 
which  represents  the  arc  whose  versed  sine  is  -  Art.  .110. 

Ex.  6.  It  is  required  to  find  the  integral  of 

x2dx 


V2ax—x'i 
Substituting,  in  Formula  c,  §  for  m,  we  obtain 

/x2dx         4a  f     x-dx         2x2     
==—  /              — = — —-V2ax—; 
V2ax—x~      3./    v2ax—xi       3 
i 
x'2dx              dx 
But 


y/2ax—xi     V2a—x 

Also 

V2a— a 


/dx  , 

z  =  -2V2a— x. 
■\l 9. n  —  f 


rr  C        x  fa  8<3     /~ %X    /TT- 

Hence  /  —  = V2a—x — —V2a—x. 

J   V2qx-x*  3  3 

(321.)  Formula  A  will  only  diminish  the  exponent  m  when 


Integration    of   Binomial   Differentials.  243 

m  is  positive  ;  but  we  may  easily  deduce  from  this  formula  an- 
other which  will  diminish  the  exponent  when  it  is  negative. 
For  this  purpose,  multiply  Formula  A  by  the  denominator 
b(np+m+l),  and  transposing  the  term  which  does  not  contain 
the  sign  of  integration,  we  obtain 

Formula  B. 
x.m-n+1  ^ + hxny+i  _  b  (np+m+l)fxm(a+bxaydx 
fxm  "(a+bxydx=-  a(m-n+l)~ 


Ex.  1.  Find  the  integral  of 
dx 


-x,orx-\l+x*)  3dx. 


;r(l+0' 

Substituting,  in  Formula  B,  —  2  for  m—n, 

1  for  a, 

1  for  6, 
3  for  n, 

and  —3-  for  p,  we  obtain 

fx-\\  +xs)~*dx^  -x~l(l  +x*f+fx(l  +x<)~hdx. 
Ex.  2.  Find  the  integral  of 

dx  -3 
3,  or  a:-2 (2— xa)  2dx. 

x*(2-xy 

Substituting,  in  Formula  B.  —2  for  m—n, 

2  for  a, 
-1  for  6, 

2  for  n, 
and  —  I  for  77,  we  obtain 

fx-\2-x*)  *dx= y—^-L—  +f(2-x*)  2dx. 

Proposition  XIV. — Theorem. 

(322.)   The  integral  of  any  differential  of  the  form 

xm(a+bxnydx, 

may  be  made  to  depend  upon  the  integral  of  another  differential 

of  the  same  form,  but  in  which  the  exponent  of  the  parenthesis  is 

diminished  by  unity. 

Let  us  put  v=x% 


244  Integral   Calculus 

where  s  is  an  exponent  to  which  any  value  may  be  assigned 
as  may  be  found  most  convenient. 

Differentiating,  we  find 

dv=sx*~1dx. 

If  then  we  assume  udv=xm(a  +  bxa)pdx,  Art.  315, 

we  must  have  u= (a+bxn)p ; 

s 

and,  by  differentiating, 

du  =  (m~S  +  1)ar~(a  +  bxydz+^z^*(a+bar)*-ldx. 
s  x  s 

But  (a+bxy=(a+bx")  (a+bxn)p-\ 

Hence 

a(m—s  +  l)+b(m—s+l+np)xn  m_s.  n.__x, 

du=— — — £J—xm  s(a+bxn)p  ldx. 

Let  the  value  of  s  be  taken  such  that 
m—  s+1  +w/?=0  ; 
that  is,  s=??i  +  l+np, 

,  —anpxm-*(a+bxy-ldx 

we  shall  have       a«= ; — . 

72^>  +  ?7Z+l 

Substituting  the  values  of  w,  v,  du,  and  rfy  here   given  u. 
formula  (1),  Art.  315,  we  obtain 

Formula  C. 

,     v    ,       xm+\a+bxy+anpfxm{a+bxn)p-idx 

Jxm  (a  +  bxn)  pdx  — j — — , 

J     v  np+m+l 

by  which  the  value  of  the  required  integral  is  made  to  depehd 
upon  another  having  the  exponent  of  the  binomial  less  by  unity. 
The  value  of  this  new  integral  may,  by  the  same  formula,  be 
made  to  depend  upon  that  of  an  integral  in  which  the  exponent 
of  the  binomial  is  still  further  diminished  ;  and  so  on  until  the 
exponent  of  the  binomial  is  reduced  to  a  fraction  less  than  unity. 

Ex.  1.  Find  the  integral  of  the  expression 
dx  vV+x2.  • 

We  may  diminish  the  exponent  of  the  binomial  by  unity  bv 
substituting,  in  Formula  C,    0  for  m, 

a3  for  a, 

1  for  6, 

2  for  n, 

£  for  p,  and  we  obtain 


Integration    of   Binomial   Differ  entials.  245 
xVcf+x"1    a~  C     dx 


fdxVaa+x*= +-J  - 


But  by  Ex.  3,  Art.  314, 
dx 


f 


Va'+x 


==log.  [x+  Vd'+X*]. 


Hence  fdxVa*+x-= +—  log.  [x+  Va3+x\l. 

Ex.  2.  Find  the  integral  of  the  expression 


dxVx^—a? 


.        xVx2— a2     a2  ,        _  . t 

Ans. —  log.  [x+  Vx2  —  a2L 

(323.)  Formula  C  will  only  diminish  the  exponent  of  the 
parenthesis  when  the  exponent  is  positive ;  but  we  may  easily 
deduce  from  this  formula  another  which  will  diminish  the  ex 
ponent  when  it  is  negative.  For  this  purpose,  multiply  Form- 
ula C  by  the  denominator  vp-\-m  +  \,  and  transposing  the  term 
which  does  not  contain  the  sign  of  integration,  we  obtain 

Formula  D. 
fx™(a+bxy->dx=  -x™+i(a+bxr+(nP+m+1)fzm("+teaYd* 

Ex.  1.  Find  the  integral  of 

(2—x2)  'dx. 

Substituting,  in  Formula  D,  0  for  m, 

2  for  a, 
-1  for  b, 
2  for  n, 
and  —  |  for  jo  — 1,  we  obtain 

f{2-xr*dx=l{2-xr\  or 


2V  2V2-X*' 

Ex.  2.  Find  the  integral  of 

xdx  _i 
■ — i,  or  x(l+xs)  3dx. 

(l+O3 

Substituting,  in  Formula  D,  1  for  m, 

1  for  a, 


246  Integral   Calculus. 

1  for  6, 
3  for  n, 
and  —  Tf  for  p—l,  we  obtain 

I  X*  2  2 

/r(l+:c3)  3dk=--(l+a:,)3+2/a:(l+a:')3<fcf 

2 

where  .r(l-f-:c3)3^£  may  be  developed  in  a  series,  and  eacr 
term  integrated  separately. 


SECTION   II. 

APPLICATIONS  OF  THE  INTEGRAL  CALCULUS-RECTIFICATION 
OF  CURVES. 

(324.)  To  rectify  a  curve  is  to  obtain  a  straight  line  equal  to 
an  arc  of  the  curve.  When  an  expression  for  the  length  oi 
a  curve  can  be  found  in  a  finite  number  of  algebraic  terms, 
the  curve  is  said  to  be  rectifiable. 

We  have  found  (Art.  251)  that  the  differential  of  an  arc  of 
a  curve,  referred  to  rectangular  co-ordinates,  is 

dz=  Vdx2+dif; 
whence  z=fVdx*+dy\  (1) 

which  is  a  general  expression  for  the  length  of  an  indefinite 
portion  of  any  curve  referred  to  rectangular  co-ordinates. 

In  order,  then,  to  rectify  a  curve  given  by  its  equation,  we 
differentiate  its  equation,  and  deduce  from  it  the  value  of  dx  oi 
dy,  which  we  substitute  in  expression  (1).  The  radical  will  then 
contain  but  one  variable,  which  being  the  differential  of  the  arc. 
its  integral  will  be  the  length  of  the  arc  itself. 

Ex.  1.  It  is  required  to  find  the  length  of  an  arc  of  the  semi- 
cubical  parabola  whose  equation  is 

if  =  a-x\ 

3 

■  y2. 

The  value  of  a;  in  this  equation  is  — . 

Differentiating  this  equation,  we  have 

i 
3y2dy  . 


dx= 


la 


and  consequently,  dx'2=^-2dy\ 

Substituting  this  value  in  the  differential  of  the  arc,  we  have 


/  9«       v  ,  ,      .      /9y+4aq         (9ij+4ay 


248  Integral   Calculus. 

Integrating  by  Art.  303,  we  have 

r/,  ,,  ,  ,    (9y+4a')f 
fVdz*+dij-= — — — +C. 

To  determine  the  constant  C,  we  see,  from  the  equation  of 
the  curve,  that  at  the  origin  of  abscissas  y=0,  and  consequently 
z=0  ;  hence 

8a2     _ 

.  whence  C  =  — — , 

and  consequently  the  entire  integral  is 

_(9y+4a2)*     So2 

Z~       27a  27' 

Ex.  2.  It  is  required  to  find  the  length  of  an  arc  of  a  circle. 
We  have  found  (Art.  227)  that  if  z  represents  an  arc  of  a 
circle,  and  t  its  tangent,  we  have 

dt        .        1 

But*  7-— -=.l-f+f-f+,  etc. 

Hence  dz=-—-2=dt-fdt+tidt-fdt-\-,  etc. 

1  +2" 

Hence,  integrating  each  term  separately,  we  obtain 

f    tb    €    f  f    t*    ft*       *     . 

/(fc«=,=,_+-__+s_,  etc.  =<(!—+_-+—,  etc.). 

If  we  take  z  equal  to  an  arc  of  30°,  its  tangent  will  be  =  </%, 
which  equals  0.577350,  which  being  substituted  for  t  in  this 
series,  we  obtain 

i  1111 

Z_^X^"^+5^    7.3'+9.84~S  GtC,) 

=  0.5235987, 

the  length  of  an  arc  of  30°,  which,  multiplied  by  6,  gives  the 

semicircumference  to  the  radius  unity 

=  3.141592. 

Ex.  3.  It  is  required  to  find  the  length  of  an  arc  of  a  cycloid 
The  differential  equation  of  the  cycloid,  Art.  275,  is 


Rectification   of  Curves. 


249 


dx— 


ydy 


dx1 


V2ry 

yW 


-If 


whence 

2ry~y 

Substituting  this  value  of  dx'  in  the  differential  of  the  arc, 

we  obtain  • 


dz=\Jt 


dy*+- 


y'dif 


2ry-y" 

,      /    2ry         T  A  /   >2r 

I         _.   ■ 

=(2r)\2r-y)  2cfy. 
Integrating  by  Art.  303,  we  obtain 

f(2r-y)Ady=-2(2r-yy+C. 

Hence  z=  -  {2r)*2  V2r-y  +  C, 

=  —  2V2r(2r—y)+C. 

If  we  estimate  the  arc  from  the 
point  B  where  y=2r,  we  shall 
have,  when  z— 0,  y=2r;  hence 

0=0+C, 
or  C  =  0, 

which  shows  that  there  is  no  constant  to  add,  and  consequent- 
ly the  entire  integral  will  be 

z  =  —  2  v,2r(2r— y), 
which  represents  the  length  of  the  arc  of  the  cycloid  from  B 
to  any  point  D  whose  co-ordinates  are  x  and  y. 

But  we  see  from  the  figure  that  BE  =  2?'— y. 

Also,         BG2=BCxBE,  Geom.,  Prop.  XXII.,  B.  IV. 
Hence  BG=  v'BCxBE=  V2v(2r-y),  _ 

and  consequently  the  arc  BD  =  2BG, 

or  the  arc  of  a  cycloid,  estimated  from  the  vertex  of  the  axis,  is 
equal  to  twice  the  corresponding  chord  of  the  generating  circle; 
hence  the  entire  arc  BDA  is  equal  to  twice  the  diameter  BC,  and 
the  entire  curve  ADBH  is  equal  to  four  times  the  diameter  of  the 
generating  circle. 

Ex.  4.  It  is  required  to  find  the  length  of  an  arc  of  the  com- 
mon parabola. 


250  Integral   Calculus. 

The  equation  of  the  parabola  is 

y*=2px. 
Differentiating,  and  dividing  by  2,  we  have  • 

ydy=pdx  : 

y~ 

whence  dx*=—dy. 

V  • 

Substituting  this  value  in  the  differential  of  the  arc,  we  obtain 

dz=\/dy*+^dy\ 

dy 


JVP*+f- 
Integrating  according  to  Ex.  1,  Art.  322,  we  obtain 


2=.v^+£Iog.(y+./7TF)+c. 

If  we  estimate  the  arc  from  the  vertex  of  the  parabola,  we 

shall  have 

?/=0  when  z=0  ; 

hence  0=|  log.  p+C,  or  C=-|  log.  p  ; 

and  consequently 


yy/p'+y*    P  locr  (y±Vf±f\ 

2p      2   °;\      p      / 

Ex.  5.  It  is  required  to  find  the  length  of  an  arc  of  the  log- 
arithmic spiral. 

The  differential  of  an  arc  of  a  polar  curve,  referred  to  polar 
co-ordinates,  Art.  257,  is 

dz=  Vdr+i-dt\ 
The  equation  df  the  logarithmic  spiral,  Art.  155,  is 
t=\og.  r. 

Udr 

Consequently  dt—- 


r 


Hence,  by  substitution,  we  find 


dz=  Vdr'+bVdr*, 


=drVl+M\ 

For  the  Naperian  system  M=l,  and  we  find 
dz=drV2; 
whence  z~rV2+C. 


Rectification    of    Curves.  251 

U  we  estimate  the  arc  from  the  pole  where  r=0,  we  have 

z  =  ?V2; 

that  is,  in  the  Naperian  logarithmic  spiral,  the  length  of  an  art 

estimated  from  the  pole  to  ant/  point  of  the  curve,  is  equal  to  the 

diagonal  of  a  square  described  on  the  radius  vector. 

Ex.  6.  It  is  required  to  find  the  length  of  an  arc  of  an  ellipse. 
The  equation  of  an  ellipse,  Art.  69,  Cor.  6,  is 

y=(l-0(A2-:r). 
Differentiating,  we  obtain 

dy=     (l-e> 
dx  y 


x  Vl—e" 

~      VA2-x2' 
Substituting  this  value  in  the  differential  of  the  arc,  we  obtain 


dz=dx\/l  + 


\A' 


A2 -or 


dx  VA'-eV 


VA2-.r2 

,  ,    /     7x* 

Mx\f  \ — rT 


VA2-.r2 
Developing  y  1 — -ry  in  a  series,  we  obu 


_.ain 

A2 

kdx  eV       eV         3eV 

The  several  terms  of  this  series  may  be  integrated  as  in 
Art.  317,  and  we  obtain 

*=x  ~h^.  -2^x  -d^6-' etc-'     (1) 

where  Xc  represents  the  arc  of  a  circle  whose  radius  is  A  and 

sine  is  x, 

A.X„    x 


„      3A2.X2    x>    .— a 

5A'.X,    x» 


X6=— -— --VA'-^etc. 

0  b 


0*0 


Integra;.   Calculus. 


In  order  to  obtain  one  fourth  of  the  circumference  of  the  el 
ripse,  we  must  integrate  between  the  limits  £=0  and  x=A 
But  when  x=A,  VA2—x2=0;  hence  the  values  of  the  quani: 
ties,  X2,  X„  etc.,  become 

A.X„ 


X2= 
X4= 


2     ' 
3Aa.X, 


Xfi= 


4 
5A2X 


3A3.X0 
2.4    ' 
3.5A\X, 


6  2.4.6 

and  consequently  equation  (1)  becomes 

e  3e4  3.3.5e 


',  etc. 


Z=X0(1-; 


;  — ,  etc.), 


2.2     2.2.4.4     2.2.4.4.6.6 

for  one  fourth  of  the  circumference  of  the  ellipse,  where  X0  is 
one  fourth  of  the  circumference  of  the  circle  whose  radius  is  A. 
Hence  the  entire  circumference  of  the  ellipse  is  equal  to 

3.3.5e6 

;  — ,  etc.) 


e-  3e4 


2.2.4.4.6.6 


QUADRATURE  OF  CURVES 

(325.)  The  quadrature  of  a  curve  is  the  measuring  of  its 
area,  or  the  finding  a  rectilinear  space  equal  to  a  proposed 
curvilinear  one.  When  the  area  of  a  curve  can  be  expressed 
in  a  finite  number  of  algebraic  terms,  the  curve  is  said  to  be 
quadrable,  and  may  be  represented  by  an  equivalent  square. 

We  have  found,  Art.  253,  that  the  differential  of  the  area  of 
a  segment  of  any  curve,  referred  to  rectangular  co-ordinates,  is 

ds=ydx, 
where  s  represents  the  area  ABPR,  and  x 
and  y  are  the  co-ordinates  of  the  point  P. 

To  apply  this  formula  to  any  particular 
curve,  we  must  find  from  the  equation  of    B 
the  curve  the  value  of  y  in  terms  of  x,  or  the 
value  of  dx  in  terms  of  y  and  dy,  and  sub- 
stitute in  the  formula  ds=ydx.     The  in-    ^~ 
tegral  of  this  expression  will  give  the  area  of  the  curve. 

Ex.  1.  It  is  required  to  find  the  area  of  the  common  parabola 

The  equation  of  the  parabola  is 
y*=2px* 


-c 


II 


X 


Quadrature    of  Curves.  253 

whence,  by  differentiating, 

,      VdlJ 
dx  = . 

P 

y'dy 

T  he  re  fore  ydx  — , 

p 

if 
and,  by  integrating,  s=— — (-C. 

If  we  estimate  the  area  from  the  vertex  of  the  parabola,  the 
constant  C  will  be  equal  to  zero,  because  when  y  is  made  equal 
to  0,  the  surface  is  equal  to  0 ;  hence  the  entire  integral  is 

■¥ 

which  equals  ^Xy2=  —  X2px=%xy  ; 

that  is,  the  area  of  a  segment  of  a  parabola  is  equal  to  two  thirds 
of  the  area  of  the  rectangle  described  on  the  abscissa  and  ordinate 
Ex.  2.  It  is  required  to  find  the  area  of  any  parabola. 
The  general  equation  of  the  parabolas,  Art.  136,  is 
y"  =  ax ; 
whence,  by  differentiating,  we  obtain 
ny"~*dy=adx, 
nyn~idy 


and  dx  — 

Therefore  ydx  = 


a 
ny*dy 


a 
And,  by  integrating,  Art.  298, 

(n+l)a 

n        V"  ^ 

or  s=— — -X  —  X*/+C, 

n  +  1      a 

ji 

= — r- r.r?/+C,  by  substituting  z  for  its 
n+1 

value  — . 
a 

If  we  estimate  the  area  from  the  vertex  of  the  parabola,  the 

constant  C  will  be  equal  to  zero ;  hence 

n 

n+1  * 

Hence  the  area  of  any  portion  of  a  parabola  is  equal  to  the 


i54 


Integral   Calculus. 


rectangle  described  on  the  abscissa  and  ordinate  multiplied  by 

7  •        n 

the  ratio 


n  +  1 


If  ?z  =  2,  the  equation  represents  the  common  parabola,  and 
the  area  equals  txl/' 

If  7i=l,  the  figure  becomes  a  triangle,  and  the  area  equals 

\*y ; 

that  is,  the  a  ea  of  a  triangle  is  equal  to  half  the  product  of  its 
base  and  perpendicular. 

Ex.  3.  It  is  required  to  find  the  area  of  a  circle. 

The  equi-aon  of  the  circle,  when  the  radius  equals  unity,  is 

i 

y=(i-xy. 

The  se-sc  ,d  member  of  this  equation  being  developed  by  the 

binomial  theorem,  we  have 

_       x1    x*     x°      5x° 

y~1~'2~~8~TG~128~>  etC' 

.               x2dx    x"dx    xBdx     5xedx 
Henc*       y*z=dz-—~ ___   etc., 

and  integrating  each  term  separately,  we  have 

.   ,  x3     xb       x1        5x° 

s=fydx=x-----—-—-,  etc.,  +C. 

D 


If  we  estimate  the  arc  from  the  point 
D,  when  x=0,  the  area  CDEH  is  0, 
and  consequently  C  =  0.  The  preced- 
ing series,  therefore,  expresses  the  area 
of  the  segment  CDEH. 

A 

If  the  arc  DE  be  taken  equal  to  30°, 


the  ^ine  of  30°,  or  its  equal  CH,  which  is  x,  becomes  =h,  and 
we  lave 

1  1 

;— ,  etc., 


CDEH=~ 


2     48     1280     1433G 
=  .4783055. 
]Rut  as  x—\,  y=  v/f ;  therefore  the  area  of  the  triangle 

CEH=lx  n/^.2165063. 
H^nce  the  area  of  the  sector 

CDE  =  .2G17992, 
wMch,  multiplied  by  12,  gives 

3.14159,  etc.,  for  the  area  of  the  whole  circle 


Quadrature   of  Curves. 


25b 


Ex   4.  It  is  required  to  find  the  area  of  an  ellipse. 
The  equation  of  the  elljpse,  referred  to  its  center  and  axes,  is 
B 


and  consequently  the  area  of  the  semi-ellipse  will  be  equal  to 
fydx=-jrfdx  VA'"-r. 


But  dxVA'—x2  is  the  differential   of  the  area  of  a  circle 
whose  radius   is  A.  Art.  254  ;   hence  the  area  of  the  ellipse 

=-r-Xthe  area  of  the  circumscribing  circle. 
A  ° 

But  the  area  of  the  circumscribing  circle  is  equal  to  ~A2 ; 

hence  the  area  of  the  ellipse  is  equal  to 

B 

or  ttAB. 

Ex.  5.  It  is  required  to  find  the  area  of  a  segment  of  an 
hyperbola. 

The  equation  of  the  hyperbola,  referred  to  its  center  and 
axes,  is 

Ay-BV=-A2B2: 

whence  y= —  Vx'  —  A3 


A 


Bdx 


ds = ydx = — r—  Vx'2  —  A'2 


Consequently 

Integrating  according  to  Ex.  2,  Art.  322,  we  obtain 


_,    vV— A"    A.B,       _         ,— — — ->     „ 

s==Bx— 2A 2"l0S*  [X+  Vx'~A  ]+    ' 

fn  order  to  determine  the  constant 
C,  make  x=A,  in  which  case  5=0, 
and  we  have 

A.B 


0  = 


log.  A+C; 


A.B 

that  is,        C=-£-  log.  A. 

Hence 


B.rV^-A2    A.B 
2A  2~ 


256 


Integral   Caiculus. 


which  represents  the  segment  APR;  hence  the  entire  segment 
APP'  is 


hxVx2-a:2 


-A.B  log. 


$x+Vx*—A*) 


which  equals 


A.B  log. 


or 


.  „  .       $Ay+Bz\ 
sy-A»Blog.{    JAB    \. 


Ex.  6.  It  is  required  to  find  the  area  of  a  cycloid. 

The  area  of  the  space  ABC  is  most 
conveniently  obtained  by  first  finding 
the  area  of  the  space  ABD,  contained 
between  the  lines  AD,  DB,  and  the 
convex  side  of  the  curve. 

LetBC  =  2r,AG=.r,FG=y;  whence 
FE=2/-— y=v.     We  shall  then  have 

d(ADEF)=ds=vdx=(2r-y)dx. 

But  the  differential  equation  of  the  cycloid,  Art.  275,  is 

ydy 


dx= — = 


y/2rv— 


y-y 


ds=dyV2ry—y-, 


Hence 

and  s  =fdy  V  2  ry — y" + C. 

.  But  this  is  evidently  the  area  of  a  segment  of  a  circle  whose 
radius  is  r  and  abscissa  y  (Art.  254)  ;  that  is,  the  area  of  the 
segment  CHI.  If  wc  estimate  the  area  of  the  first  segment 
ADEF  from  AD,  and  the  area  of  the  segment  CHI  from  the 
point  C,  they  will  both  be  0  when  y=0 ;  the  constant  C,  to  be 
added  in  each  case,  will  then  be  0,  and  we  shall  have 

ADEF=CHI; 
and  when  y  =  2r, 

ADB=the  semicircle  CHB= — . 

2 

But  the  area  of  the  rectangle  ADBC  is  equal  to 

ACxAD  =  7rrX2/~27T;-\ 

Hence  the  area  AFBC=ADBC  —  ADB  =  |7rr"= three  times  the 

semicircle  CHB  ;  and  doubling  this,  we  find  the  area  included 

between  one  branch  of  the  cycloid  and  its  base,  is  equal  to  three 

times  the  area  of  the  generating  circle. 


Area   of  Spirals. 


257 


ds—- 


AREA  OF  SPIRALS. 

(32G.)  The  differential  of  the  area  of  a  segment  of  a  polar 
curve.  Art.  258,  is 

2  ' 

Ex   1.  It  is  required  to  find  the  area  of  the  spiral  of  Archi- 
medes. 

The  equation  of  the  spiral  of  Archimedes,  Art  148,  is 

t 
2r7; 

dt 

'2r? 
ds=TTr"~dj- ; 
Trr3        f 

If  we  make  t=2n,  we  have 


r=; 


whence 


hence 


dr 


247T2' 


s= 


3' 


which  is  the  area  PMA  described  by  one 

revolution  of  the  radius  vector.     Hence  3VL 

the  area  included  by  the  first  spire  is  equal  to  one  third  the 

area  of  the  circle,  whose  radius  is  equal  to  the  radius  vector 

after  the  first  revolution. 

If  we  make  £=2(2~),  we  have 

8tt 


s= 


3' 


which  is  the  whole  area  described  by  the  radius  vector  during 
two  revolutions.  But  in  the  second  rev- 
olution, the  radius  vector  describes  the 
part  PMA  a  second  time  ;  hence,  to  ob- 
tain the  area  PNB,  we  must  subtract 
that  described  during  the  first  revolution  ; 
hence 

the  area  PNB= =  — , 

3      3      3 

Ex.  2.  It  is  required  to  find  the  area  of  the  hyperbolic  spiral. 
The  equation  of  the  hyperbolic  spiral,  Art.  151,  is 

R 


258  Integral   Calculus. 

a 
r=_. 

,      a*dt 
whence  "2?"' 

a2 
and  s=--. 

Ex.  3.  It  is  required  to  find  the  area  of  the  logarithmic 

spiral. 

The  equation  of  the  logarithmic  spiral,  Art.  155,  is 

t—\og.  r. 

J     Mdr 

blence  dt= . 

r 

rdr 
When  M=l,  ds=—, 

<& 

T 

and  s=— +C. 

4 

If  we  estimate  the  area  from  the  pole  where  r=0  and  C=0 

we  have 

ra 

S=4"; 
that  is,  the  area  of  the  Naperian  logarithmic  spiral  is  equal  to 
one  fourth  the  square  described  upon  the  radius  vector. 

AREA  OF  SURFACES  OF  REVOLUTION. 

(327.)  We  have  found  (Art.  255)  that  the  differential  of  the 
area  of  a  surface  of  revolution  is 


dS  =  2nyVdx,i  +  dy2; 
whence  S=f2ny  Vdz"+dy*,  (1) 

which  is  a  general  expression  for  the  area  of  an  indefinite  por- 
tion of  a  surface  of  revolution  ;  the  axis  of  X  being  the  axis 
of  revolution,  and  Vdx^+dy"2  the  differential  of  the  arc  of  the 
generating  curve. 

In  order  to  obtain  the  area  of  any  particular  surface,  we 
differentiate  the  equation  of  the  generating  curve,  and  deduce 
from  it  the  values  of  y  and  dy  in  terms  of  £  and  dx ;  or  of  dx 
in  terms  of  y  and  dy,  which  we  substitute  in  expression  (1). 
The  integral  of  this  expression  will  be  the  area  required. 

Ex.  1.  It  is  required  to  determine  the  convex  surface  of  a 
cone. 


Area   of   Surfaces    of   Revolution. 


259 


B 


If  the  right-angled  triangle  ABC  be  re- 
solved about  AB,  the  hypothenuse  AC 
will  describe  the  convex  surface  of  a 
cone.  Let  AB=A,  BC=b,  and  let  x  and 
y  be  the  co-ordinates  of  any  point  of  the 
line  AC,  referred  to  the  point  A  as  an 
origin  ;  we  shall  then  have 

x  :  y  : :  h  :  b  ; 
bx 
whence  #=T" 

By  differentiation,  we  obtain 

dy=jdx,  and  dy2=-j~idx'2. 

Substituting  these  values  of  y  and  dy"2  in  the  general  formula, 
we  have 

bx 


f2ny  V  dx2 + dy'  =/2n-^dx  V  1? + b% 


bx' 


=  n^rVk*+b*+C. 

If  we  estimate  the  area  from  the  vertex  where  x=0,  we 
have  C  =  0,  and 

bx' 


S=TT—Vh2+b\ 
If 

Making  x=AB=h,  we  have  the  surface  of  the  cone  whose 
altitude  is  h,  and  the  radius  of  its  base  b, 

7vbVfii+b2=2Trbx—; 

that  is,  the  convex  surface  of  a  cone  is  equal  to  the  circumference 
of  its  base  into  half  its  side. 

Ex.  2.  It  is  required  to  determine  the  convex  surface  of  a 
cylinder. 

If  the  rectangle  ABCD  be  revolved  about 
the  side  AB,  the  side  CD  will  describe  the 
convex  surface  of  a  cylinder.  Let  AB=h, 
and  CA  =  &;  the  equation  of  the  straight 
line  CD  will  be  y=b ;  whence 

dy=0. 

Substituting  these  values  in  the  general  formula,  we  obtain 


D 


260 


Integral   Calculus. 


f2ny  Vdz*+dy*=f2nbdz, 

=  2nbx+C. 
It  we  estimate  the  area  from  the  point  A  where  x=0,  C  be- 
comes equal  to  0;  and  if  we  make  x—AR  —  h,  we  have  the 
convex  surface  of  the  cylinder 

2nbh ; 
that  is,  the  convex  surface  of  a  cylinder  is  equal  to  the  circum- 
ference of  its  base  into  its  altitude. 

Ex.  3.  It  is  required  to  determine  the  surface  of  a  sphere. 

The  equation  of  the  generating  circle,  referred  to  the  center 

as  an  origin,  is 

x*+y*=R\ 

By  differentiating,  we  obtain 

xdx-\-ydy=0  ; 

.  xdx 

dy=  — 


whence 
and 


dxf 


y 

x'dx* 


Substituting  this  value  in  the  general  formula,  we  obtain 
f27ry\/(^+l)dx°=f2TTdx  vV+7", 


=f2nRdx, 
=  2ttRx+C. 


0) 


To  determine  the  constant,  we  will  sup- 
pose the  integral  to  commence  at  the  center 
of  the  sphere  ;  and  since  the  origin  of  co-or- 
dinates is  at  the  center,  the  integral  will  be 
zero  when  x=0,  and  therefore  the  constant 
is  equal  to  zero.  Making  .r=R,  we  have 
for  the  surface  of  a  hemisphere 

2;rR2, 

and  theiefore  the  surface  of  the  sphere  is 

4:rR2; 

that  is,  the  surface  of  a  sphere  is  equal  to  four  of  its  great  circles. 

Ex.  4.  It  is  required  to  determine  the  surface  of  a  parabo* 
bid. 


Area   of   Surfaces   of   Revolution  201 

A  paraboloid  is  a  solid  described  by  the  revolution  of  an  arc 
AC  of  a  parabola  about  its  axis  AB. 

The  equation  of  the  parabola  is 
y*=2px, 
which,  being  differentiated,  gives 

,     ydy      ,  ,  a   yW 

dx= -,  and  ax  —■ 


V  P 

Substituting  this  value  in  the  general  formula,  it  reduces  ko 

dS=2ny\/(fS^  di/Jjydy  Vf+f. 

Integrating  according  to  Art.  303,  we  obtain 
2tt  2 

To  determine  the  constant  C,  let  us  suppose  that  y  becomes 
zero,  in  which  case  S  also  reduces  to  zero,  and  the  preceding 
equation  becomes 

0=^-+C; 

2rrp* 
whence  C=  — 


3    ' 

and  supposing  the  integral  to  be  taken  between  the  limits  y=0 
and  y—b,  the  entire  integral  will  be 

Ex.  5.  It  is  required  to  determine  the  surface  of  an  ellipsoid. 

described  by  revolving  an  ellipse  about  its  major  axis. 

According  to  Art.  255,  we  have 

dS  =  2rryVdx'  +  dy% 

or  dS=2Trydz. 

But  in  Ex.  0,  Art.  324,  we  have  found 

kdx  eV      e*x*        3eV 

dZ"  V^r^x~i{1~2A~2AA~2AlAl     '  etC°  ; 

2-Aydx  eV      e*x*         Seex6 

hence   ^=-^==(i_— -^^-— x.-,  etc.,. 

But  iL.=B. 

VA'-x3 

e*x*      e*x*         3e6x6 
Hence  dS  ^mx{\-^-^^-—^--,  etc.), 


262  Integral   Calculus. 

and  integrating  each  term  separately,  we  obtain 

Integrating  between  the  limits  x=0  and  x—  A,  we  shall  ob- 
tain half  the  surface  of  the  ellipsoid 

e2         e4  3e6 

=  2^(1---—-^^-,  etc.), 

or  the  entire  surface  of  the  ellipsoid  equals 
e2         e4  3e6 

4ffAB(i___i— _s_.lrt0.). 

Ex.  6.  It  is  required  to  determine  the  surface  described  by 
the  revolution  of  a  cycloid  about  its  base. 

The  general  formula  for  the  differential  of  the  surface  is 

dS=2nydz. 
But  we  have  found  in  Ex.  3,  Art.  324. 


dz 


J  v  2ry—y 


Hence 


dS—2nydy 


2TrV2iy2dy 
2ry-yz      V2ry-tf  ' 


which,  being  integrated,  will  give  the  value  of  the  surface  re- 
quired. 

But,  according  to  Ex.  6,  Art.  320, 


/ 


Hence 


irdy  8r    .—— 

V2ry—y2         3 


2y 


y~~t^2r~y- 


f 


2nV2ry*dy     n      .-— r     8r       2y       -, 

—  2n  V2r[— —  V2r— y — —  V2r— y]+G. 


V2ry— y2 
If  we  estimate  the  surface  from 
the  plane  passing  through  B,  we 
shall  have  S  =  0  when  y=2r,  and 
consequently  C  =  0.  If  we  then 
integrate  between  the  limits 

y=0  and  y=2r, 
we  have  half  the  surface=3327rr2 ; 

hence  the  entire  surface  =  6g-47rr!l ; 

that  is,  the  surface  described  by  the  cycloid  revolved  about  its 
base,  is  equal  to  G4  thuds  of  the  generating  circle. 


CUBATUSE     OF     SoLIDS     OF     REVOLUTION.  263 

CUBATURE  OF  SOLIDS  OF  REVOLUTION. 

(328.)  The  cubature  of  a  solid  is  the  finding  its  solid  con- 
tents, or  finding  a  cube  to  which  it  is  equal. 

We  have  found,  Art.  256,  that  the  differential  of  a  solid  oi 
revolution  is 

dV=7ry*dx ; 
whence  Y=fny2dx,  (1 ) 

where  x  and  y  represent  the  co-ordinates  of  the  curve  which 
generates  the  bounding  surface,  the  axis  of  X  being  the  axis 
of  revolution. 

For  the  cubature  of  any  particular  solid,  we  differentiate  the 
equation  of  the  generating  curve,  and  deduce  from  it  the  value 
of  dx  in  terms  of  y  and  dy,  or  the  value  of  y2,  in  terms  of  x 
which  we  substitute  in  expression  (1).  The  integral  of  this 
expression  will  be  the  solid  required. 

Ex.  1.  It  is  required  to  determine  the  solidity  of  a  cylinder. 

Let  b  represent  AC,  the  radius  of  the  base,   c p 

and  h  the  altitude  AB.     Then 

V  =JTx\fdx  —Jidfdx, 

=  nb\x  +  C.  A  ~~B 

Taking  the  integral  between  the  limits  x=0  and  x=AB=h. 
we  have 

V=nb*h; 
that  is,  the  solidity  of  a  cylinder  is  equal  to  the  product  of  its 
base  by  its  altitude. 

Ex.  2.  It  is  required  to  determine  the  solidity  of  a  cone. 
Let  h  represent  the  altitude  of  the  cone,  and  r  the  radius  of 
its  base.     We  shall  then  have,  by  Ex.  1,  Art.  327, 

?/--x,  and  y'=pc\ 

Substituting  this  value  of  y1  in  the  general  formula,  it  be- 
comes 

r8 

dY~—nx2dx; 
h' 

r'lrx3 
whence  V=  „.,.  +C. 

3/r 

And  taking  the  integral  between  the  limits  .r=0  and  x=h,  we 
obtain 


264  Integral   Calculus. 

h 

V=i7rr2A=7rraX-; 

o 

that  is,  the  solidity  of  a  cone  is  equal  to  the  area  of  its  base  into 
one  third  of  its  altitudi. 

Ex.  3.  It  is  required  to  find  the  solidity  of  a  prolate  spheroid, 
or  the  solid  described  by  the  revolution  of  an  ellipse  about  its 
major  axis. 

The  equation  of  an  ellipse  is 

Substituting  this  value  of  y2  in  the  general  formula,  it  be- 
comes 

B2 
dV=7r—  (A*-x*)dz, 

and  by  integrating,  we  find 


B2  x*\ 

V=.-(A'*--)+C. 


If  we  estimate  the  solidity  from  the  plane  passing  through 
the  center  perpendicular  to  the  major 
axis,  we  shall  have  when  x—0,  V=0, 
and  consequently  C— 0.     Therefore 

Making  x=A,  we  obtain  for  one  half  of  the  spheroid 
frrB'A; 
and  consequently  the  entire  spheroid  equals 
|ttB2A,  or  |7tB2X2A. 
But  ttB2  represents  the  area  of  a  circle  described  upon  the 
minor  axis,  and  2A  is  the  major  axis  ;  hence  the  solidity  of  a 
prolate  spheroid  is  equal  to  two  thirds  of  the  circu??iscribing 
cylinder. 

Cor.  If  we  make  A=B,  we  obtain  the  solidity  of  the  sphere 

f7rR3=i7rD\ 
Ex.  4.  It  is  required  to  find  the  solidity  of  the  common  pa 
raboloid. 

The  equation  of  the  parabola  is 
y*=2px. 


CuJature  of    Solids    of   Revolution.         2G5 

Substitut/.ng  this  value  of  y1  in  the  general  formula,  it  be- 
comes 

dV=2npxdx. 
Hence  V==7rpa:a+C. 

To  determine  the  constant,  we  suppose  x  to  become  equal  to 
zero,  in  which  case  the  solidity  is  zero,  and  C  =  0. 

Taking  the  integral  between  the  limits  x—0  and  z—h,  and 
designating  by  b,  the  ordinate  corresponding  to  the  abscissa 
x  =  h,  we  have 

/St 

But  7r&a  represents  the  area  of  a  circle 
of  which  BC  is  the  radius ;  hence  the 
solidity  of  the  paraboloid  is  one  half  that 
of  the  circumscribed  cylinder. 

Ex.  5.  It  is  required  to  find  the  solidify  of  the  solid  generated 
by  the  revolution  of  any  parabola  about  its  axis. 
The  general  equation  of  the  parabolas  is 
yn=ax  ; 

whence  dz=— -, 

a 

nnyn+1dy 


and  dV-- 


a 

,n+2 


Hence  V=rSjr+C' 

(n+2)a 


71  o"    V        « 

Xtti/2X-+C, 


w+2       J       a 
n 


-Try-x  +  C. 

71  +  2    J 

But  when  .r=0,  V=0,  and  therefore  C=0 ;  hence 

V= — ; — "ny  x. 
n+2  J 

If  n—2,  the  solid  becomes  the  common  paraboloid,  and  its 
solidity  equals  \^ifx. 

If  n=l,  the  curve  becomes  a  straight  line,  and  the  solid  be- 
comes a  cone,  and  its  solidity  equals  \ny*x. 

Ex.  6.  It  is  required  to  find  the  solidity  of  the  solid  gener- 
ated by  the  revolution  of  the  cycloid  about  its  base. 


266  Integral   Calculus. 

The  general  formula  for  the  differential  of  a  solid  of  revoiu 
tion  is 

dV=Try*dx. 
But  we  have  found  for  the  cycloid,  Art.  275, 

ydy 


dx—- 


Hence  dV= 


y/2ry—ya 

rry'dy 


V2ry—y* 

which,  being  integrated,  will  give  the  value  of  the  solid  re- 
quired. 

The  integral  of  this  expression  has  already  been  found  in 
Art.  319. 


Hence  Y=Tr(-^-X2-j-V2ry-y*), 


3?v    y 


where  yii=— Xt—  -  y/2?-y— y1, 


X!=X0—  V2ry—y% 
and  X0=  arc  of  which  r  is  the  radius  and  y  the 

versed  sine. 

We  must  now  integrate  between  the  limits  y=0  and  y=2r. 

When  3/=0,  all  the  above  terms  become  0. 

When  y=2r,  these  values  become 

X0=77r, 

X1=X0=7rr, 
_3r     _37rra 

e*      2     3 

and  Y=^f, 

which  is  one  half  of  the  solidity ;  hence  5ttVs  is  the  solid  re- 
quired. 

But  7r(2r)2  represents  the  base  of  the  circumscribing  cylinder, 
2nr  represents  its  altitude, 
and      87rV3  represents  its  solidity. 

Hence  the  solid  required  is  equal  to  Jive  eighths  of  the  cir- 
cumscribing cylinder. 


EXAMPLES    FOR    PRACTICE. 

ANALYTICAL  GEOMETRY. 
Application  of  Algebra  to  Geometrij. 
(329.)  Ex.  1.  In  a  right-angled  triangle,  having  given  the  hy- 
pothenuse  (a),  and  the  difference  between  the  base  and  perpen- 
dicular (2d),  to  determine  the  two  sides 


AM.s/£^+a,a*\/?^~d. 


Ex.  2.  Having  given  the  area  (c)  of  a  rectangle  inscribed  in 
a  triangle  whose  base  is  (6)  and  altitude  (a),  to  determine  the 
height  of  the  rectangle. 


a     .  fa1 ac 


2~v    4 

Ex.  3.  Having  given  the  ratio  of  the  two  sides  of  a  triangle, 
as  m  to  n,  together  with  the  segments  of  the  base,  a  and  b,  made 
by  a  perpendicular  from  the  vertical  angle,  to  determine  the 
sides  of  the  triangle. 


t/a2-6a         ,     ./or 


Ex.  4.  Having  given  the  base  of  a  triangle  (2a),  the  sum  of 
the  other  two  sides  (2s),  and  the  line  (c)  drawn  from  the  ver- 
tical angle  to  the  middle  of  the  base,  to  find  the  sides  of  the 

triangle.  

Ans .  s±  Va"  +  <r  —  s2 . 

Ex.  5.  Having  given  the  two  sides  (a)  and  (b)  about  the  ver- 
tical angle  of  a  triangle,  together  with  the  line  (c)  bisecting  that 
angle  and  terminating  in  the  base,  to  find  the  segments  of  the 
base. 

Ans.  V^T->  and  bV  ^W" 

Ex.  6.  Determine  the  sides  of  a  right-angled  triangle,  having 
given  its  perimeter  (2p),  and  the  radius  (r)  of  the  inscribed  circle. 
Ans.  The  hypothenuse  equals  p  —  r,  and  the  other  sides  are 
p  +  r  ±  V{p  —  r  f  —  4pr 
2  ' 


268  Examples    for    Practice. 

Ex.  7.  The  area  of  an  isosceles  triangle  is  equal  to  a,  and 
each  of  the  equal  sides  is  equal  to  c.  What  is  the  length  of 
the  base  ? 

Ans.  v/2c2±2\/ci  —  4a2. 

Ex.  8.  The  area  of  an  isosceles  triangle  is  100  square  inches, 
and  each  of  the  equal  sides  is  20  inches.  What  is  the  length 
of  the  base  ? 

Ans.  38.637  or  10.356. 

Ex.  9.  The  sum  of  the  two  legs  of  a  right-angled  triangle  is 
s,  and  the  perpendicular  let  fall  from  the  right  angle  upon  the 
hypothenuse  is  a.     What  is  the  hypothenuse  of  the  triangle  ? 

Ans.  Vs2  +  a2—a. 
Ex.  10.  The  area  of  a  right-angled  triangle  is  equal  to  a,  and 
the  hypothenuse  is  equal  to  h.     What  are  the  two  legs  ? 

Ans.  lVh2  +  4a±±V]r  —  4a. 

Ex.  1 1.  From  two  points,  A  and  B,  both  situated  on  the  same 
side  of  a  straight  line,  the  perpendiculars  AC  =  b  and  BD  —  a 
are  let  fall  upon  this  line,  and  the  distance,  CD,  between  their 
points  of  intersection  is  equal  to  c.  At  what  distance  from  C 
in  the  given  straight  line  must  the  point  F  be  taken,  so  that  the 
straight  lines  AF  and  BF  may  be  equal  to  each  other  ? 

,,         r.-n     a2  +  c2  —  b2 
Ans.  CF  =  — ■ . 

Ex.  12.  The  same  construction  remaining  as  in  the  preceding 
problem,  where  must  the  point  F  be  taken  in  the  given  straight 
line,  so  that  the  angle  AFC  may  be  equal  to  the  angle  BFD  ? 

Ans.  CF=-^-. 
a+b 

Ex.  13.  At  what  distance  from  C  in  the  given  straight  line 
must  the  point  F  be  taken,  so  that  the  two  triangles  ACF  and 
BDF  may  contain  equal  areas  ? 

Arts.  CF=-^7. 
a  +  b 

Ex.  14.  At  what  distance  from  C  must  the  point  F  be  taken, 
so  that  the  area  of  the  triangle  ABF  may  be  equal  to  d  ? 

*  /^,Ti        *CCt  —  DC 

Ans.  CF= j-. 

a—b 


Examples    for   Practice.  269 

Ex.  15.  At  what  distance  from  C  must  the  point  F  be  sit- 
uated, so  that  the  angle  AFB  may  be  a  right  angle  ? 

Ans.  CF  =  !±£Vca-4a6. 

Ex.  lG.  At  what  distance  from  C  must  the  point  F  be  sit- 
uated, so  that  the  circle  passing  through  the  points  A,  F,  and  B 
may  touch  the  given  straight  line  in  the  point  F  ? 

.  -  be  ±  Vab[c-  +  (a~bf] 

Ans.  Lr  = —. -. 

a  —  b 

Ex.  17.  At  what  distance  from  C  must  the  point  F  be  sit- 
uated, so  that  AF  may  have  to  BF  the  ratio  of  n  to  m  ? 

_,  _,     —  n2c  ±  Vm2n2(  cr  +  b~  +  c2 )  —  n*a2 — m%U2 

Ans.  CF=r — 5 5 — , 

nr  —  n* 

Ex.  18.  One  of  the  angular  points  of  an  equilateral  triangle 
falls  on  the  angle  of  a  square  whose  side  is  a,  and  the  other  an- 
gular points  lie  on  the  opposite  sides  of  the  square.  What  is 
the  length  of  a  side  of  the  triangle,  and  what  is  its  area  ? 

Ans.  The  side  of  the  triangle  is  a(  -\/6—  -v/2), 
and  its  area  is  a\2\/3  —  3). 
Ex.  19.  The  area  of  a  ring  contained  between  two  concentric 
circles  is  a,  and  its  breadth  is  b.     What  are  the  radii  of  the 

circles  ? 

a       b       .    a      b 

Ans.  —. and  -r — \--. 

2bn     2  2bn     2 

Ex.  20.  Determine  the  radii  of  three  equal  circles,  described 
in  a  given  circle,  which  touch  each  other,  and  also  the  circum- 
ference of  the  given  circle  whose  radius  is  R. 

Ans.  R(2l/3-3). 

Ex.  21.  Having  given  the  three  lines  a,  b,  and  c,  drawn  from 
the  three  angles  of  a  triangle  to  the  middle  of  the  opposite  sides, 

to  determine  the  sides.  

Ans.  %y/2a2  +  2b*-c2, 
|V2a2  +  2c2~-/r, 
|V262  +  2c2— cr. 

Ex.  22.  Having  given  the  hypothenuse  (a)  of  a  right-angled 
triangle,  and  the  radius  (r)  of  the  inscribed  circle,  to  determine 
the  other  sides. 

Ans.  l{a  +  2r±  Vcr-4.ar— 4r2). 


270  Examples    for   Practice. 

The  Straight  Line. 
Ex.  23.  Construct  the  line  whose  equation  is 

Ex.  24.  Construct  the  line  whose  equation  is 
y2  —  5x2. 

Ex.  25.  Find  the  lengths  of  the  sides  of  a  triangle,  the  co- 
ordinates of  whose  vertices  are 

a/=2,  y'=3;  x"  =  4,  y"  =  —  5  ;  oc"'=—  3,  y///=—  6, 
the  axes  being  rectangular. 

Ans.  V68,  ^50,  V10^. 

Ex.  26.  Find  the  co-ordinates  of  the  middle  points  of  the 
sides  of  the  triangle,  the  co-ordinates  of  whose  vertices  are 
(4,  6),  (8,  -10),  (-6,  -12). 

Ans.{Q,  -2),  (1,  -11), (-1,  _3). 

Ex.  27.  The  line  joining  the  points  whose  co-ordinates  are 
(6,  9),  (12.  —15)  is  trisected.  Find  the  co-ordinates  of  the 
point  of  trisection  nearest  the  former  point. 

Ans.  x=8,  y=l. 

Ex.  28.  The  co-ordinates  of  a  line  satisfy  the  equation 
x2+y2  —  4x—6y=l8. 
What  will  this  equation  become  if  the  origin  be  removed  to  the 
point  whose  co-ordinates  are  (2,  3)  ? 

Ans.  X2  +  Y2  =  31. 

Ex.  29.  The  co-ordinates  of  a  line,  when  referred  to  one  set 
of  rectangular  axes,  satisfy  the  equation 

y2—x2  =  6. 
What  will  this  equation  become  if  referred  to  axes  bisecting 
the  angles  between  the  given  axes  ? 

Ans.  XY  =  3. 

Ex.  30.  What  points  are  represented  by  the  two  equations 

x2-\-y2  —  2b,  and  x— y—\  ? 

Ans.  (4,  3),  (-3,  -4). 

Ex.  31.  The  equation  of  a  line  referred  to  rectangular  axes  is 
3x+4y  +  20  =  0. 
Find  the  length  of  the  perpendicular  let  fall  upon  it  from  the 
origin. 

Ans.  4. 


Examples    for   Practice.  271 

Ex.  32.  The  equation  of  a  line  referred  to  rectangular  axes  is 
2x+y  —  4  =  0.' 
Find  the  length  of  the  perpendicular  let  fall  upon  it  from  the 
point  whose  co-ordinates  are  (2,  3). 

a  3 

Ans.  —-. 

V5 
Ex.  33.  Form  the  equations  of  the  sides  of  a  triangle,  the  co- 
ordinates of  whose  vertices  are  (2,  1),  (3,  —2),  (  —  4,  —1). 
Ans.  3a?+y=7;  a?+7z/+ll  =  0;  3y-a:=l. 

Ex.  34.  Form  the  equations  of  the  sides  of  the  triangle,  the 
co-ordinates  of  whose  vertices  are  (2,  3),  (4,  —5),  (  —  3,  —6). 
Ans.  <lx+y=  11;  x— 7?/  =  39;  9a:—  5y=3. 

Ex.  35.  Find  the  area  of  the  triangle,  the  co-ordinates  of 
whose  vertices  are  (2,  1),  (3,  —2),  (  —  4,  —1). 

Ans.  10. 

Ex.  36.  Find  the  area  of  the  triangle,  the  co-ordinates  of 
whose  vertices  are  (2,  3),  (4,  —5),  (  —  3,  —6). 

Ans.  29. 

Ex.  37.  Prove  that  the  three  perpendiculars  drawn  from  the 
angles  to  the  opposite  sides  of  a  triangle,  pass  through  the  same 
point. 

Ex.  38.  Prove  that  the  three  straight  lines  drawn  from  the 
angles  of  a  triangle  to  bisect  the  opposite  sides,  pass  through  the 
same  point. 

The  Circle. 
Ex.  39.  Find  the  co-ordinates  of  the  centre,  and  the  radius 
of  the  circle  whose  equation  is 

x2  +  y2— 4a?+4?/  =  8. 
Ans.  Radius  =  4,  co-ordinates  of  centre  (2,  —2). 

Ex.  40 .  Find  the  co-ordinates  of  the  centre,  and  the  radius 
of  the  circle  whose  equation  is 

x2  +  y2  +  8x-6y=\l. 
Ans.  Radius  =  0,  co-ordinates  of  centre  (  —  4,  3). 

Ex.  41.  Find  the  co-ordinates  of  the  centre,  and  the  radius 
of  the  circle  whose  equation  is 

x2  +  y2  —  2x—4:i/  =  20. 
Ans.  Radius  =  5,  co-ordinates  of  centre  (1,  2). 


272  Examples    for    Practice 

Ex.  42.  Find  the  co-ordinates  of  the  points  in  wnich  the  cir- 
cle whose  equation  is  x2+y2  =  65,  is  intersected  by  the  line 
whose  equation  is  3x+y  =  25. 

Ans.  (7,  4)  and  (8,  1). 
Ex.  43.  Find  the  points  where  the  axes  are  cut  by  the  circle 
whose  equation  is  x2-\-y2  —  5a:— 7?/  + 6  =  0. 

Ans.  x=3,  oc=2;  y=zQ,y=l. 
Ex.  44.  Find  the  co-ordinates  of  the  centre,  and  the  radius 
of  the  circle  whose  equation  is 

y2-\-x2—  I0?j  —  5x-\-l=0. 
Ans.  Radius  =  5.5;  co-ordinates  of  centre  (2-^,  5). 
Ex.  45.  Find  the  point  on  the  circumference  of  a  circle  whose 
radius  is  6  inches,  from  which  if  a  radius  and  a  tangent  line  be 
drawn,  they  will  form,  with  the  axis  of  X,  a  triangle  whose  area 
is  30  inches  ? 

Ans.  o:= 3.0870;  y  =  5.1449. 
Ex.  46.  Find  the  co-ordinates  of  the  points  in  which  the  cir- 
cle whose  equation  is  y2  + x2—  lOy  —  5x+ 1  =0,  is  intersected  by 
the  line  whose  equation  is  y  =  3  +  5x. 

Ans.  x=  1.4825  or  —0.5209;  y=  10.4125  or  0.3955. 
Ex.  47.  Find  the  co-ordinates  a,  b,  a',  b' ,  of  the  centres,  and 
the  radii  r,  r',  of  the  two  circles  y2  +  x2  —  20a?— 40  =  0,  and 
y2-\rx2— 40y  +  50  =  0 ;  and  in  case  the  two  circumferences  cut 
each  other,  what  are  the  co-ordinates  of  the  points  of  intersec- 
tion ? 

Ans.  a  =  10,  5  =  0,  r— 11.8321,  a'  =  0,  ^  =  20,  r'=  18.7082, 
a:=  15.9577  or  -1.7577,  ?/=  10.2260  or  1.3740. 

The  Parabola. 
Ex.  48.  The  equation  of  a  parabola  is  ?/2  =  4a\     What  is  the 
abscissa  corresponding  to  the  ordinate  7  ? 

Ans.  12|. 
Ex.  49.  The  equation  of  a  parabola  is  y2—\Sx.     What  is  the 
ordinate  corresponding  to  the  abscissa  7  ? 

Ans.  ±^/126. 
Ex.  50.  On  a  parabola  whose  equation  is  y2=\0x,  a  tangent 
line  is  drawn  through  the  point  whose  ordinate  is  8.     Determ- 
ine where  the  tangent  line  meets  the  two  axes  of  reference. 
Ans.  Distance  on  X  =  6.4;  distance  on  Y  =  4. 


Examples   for   Practice.  273 

Ex.  51.  On  a  parabola  whose  equation  is  y2—12x,  find  the 
point  from  which  if  a  tangent  and  normal  be  drawn,  they  will 
form  with  the  axis  of  X  a  triangle  whose  area  is  35. 

Ans.  ^  =  2.9164;  y  =  5.9158. 
Ex.  52.  Find  the  co-ordinates  of  the  points  in  which  the  pa- 
rabola whose  equation  is  y2—4x  is  intersected  by  the  line  whose 
equation  is  y  =  2x—  5. 

Ans.  ?/  =  4.3166  or  —2.3166;  a?=4.6583  or  1.3417. 
Ex.  53.  Find  the  co-ordinates  of  the  points  in  which  the  pa- 
rabola whose   equation  is  y2—  18a?  is   intersected  by  the  line 
whose  equation  is  y  =  2x—5. 

Ans.y  =  12.5777  or  —3.5777;  cr=8.7888  or  0.711 1. 
Ex.  54.  Find  the  co-ordinates  of  the  points  in  which  the  pa- 
rabola whose   equation  is  ?/3  =  4a?  is  intersected  by  the  circle 
whose  equation  is  ,r2  +  ?/2=:64. 

Ans.  .r=z6.2462;  y=±4.9985. 
Ex.  55.  Find  the  co-ordinates  of  the  points  in  which  the  pa- 
rabola whose  equation  is  y2=l8x  is  intersected  by  the  circle 
whose  equation  is  ar  +  ?/2  =  64. 

A?is.  x- 3. 0416;  y—  ±7.3992. 

The  Ellipse. 
Ex.  56.  In  an  ellipse  whose  major  axis  is  50  inches,  the  ab- 
scissa of  a  certain  point  is  15  inches,  and  the  ordinate  16  inches, 
the  origin  being  at  the  centre.     Determine  where  the  tangent 
passing  through  this  point  meets  the  two  axes  produced. 

Ans.  Distance  from  the  centre  on  the  axis  of  X  =  41§, 
on  the  axis  of  Y  =  25  inches. 
Ex.  57.  Find  the  angle  which  the  tangent  line  in  the  pre- 
ceding example  makes  with  the  axis  of  X.  Ans.  30°  57'. 

Ex.  58.  Find  where  the  normal  line  passing  through  the  same 
point  as  in  the  preceding  example  meets  the  two  axes. 

Ans.  Distance  from  centre  on  axis  of  X  =  5-§; 
on  the  axis  of  Y  =  9  inches. 
Ex.  59.  Find  the  point  on  the  curve  of  an  ellipse  whose  two 
axes  are  50  and  40  inches,  from  which  if  an  ordinate  and  nor- 
mal be  drawn,  they  will  form  with  the  major  axis  a  triangle 
whose  area  is  80  inches. 

Ans.  *=17.6777;  ?/=14.1421  inches. 
'    S 


274  Examples   for  Practice. 

Ex.  60.    On   an    ellipse    whose    two    axes    are    50   and   40 

inches,  find  the  point  from  which  a  tangent  line  must  be  drawn 

in  order  that  it  may  make  an  angle  of  35  degrees  with  the 

axis  of  X. 

Ans.  07=16.4656  ;  y— 15.0495  inches. 

Ex.  61.  The  axes  of  an  ellipse  are  50  and  40  inches,  and  the 

variable  angle  is  36  degrees,  the  pole  being  at  one  of  the  foci. 

Determine  the  radius  vector. 

Ans.  10.771  inches. 

Ex.  62.  The  axes  of  an  ellipse  are  50  and  40  inches,  and  the 
radius  vector  is  12  inches.     Determine  the  variable  angle. 

Ans.  56°  W. 

Ex.  63.  In  an  ellipse  whose  major  axis  is  50  inches,  the  ra- 
dius vector  is  12  inches,  and  the  variable  angle  is  36  degrees, 
determine  the  minor  axis  of  the  ellipse. 

Ans.  42.47  inches. 

Ex.  64.  Find  the  co-ordinates  of  the  points  in  which  the  el- 
lipse whose  equation  is  2by2  + 1  Gx2  =  400,  is  intersected  by  the 
line  whose  equation  is  y  =  2x  —  5. 

Ans.  a?=  +  3.7998  or  +0-5104;  y=  +2.5996  or  -3.9792. 

DIFFERENTIAL    CALCULUS. 
Differentials  of  Functions. 

Ex.  1 .  What  is  the  differential  of  the  function 

u — x{a  +  x)(  a2  +  x2 )  ? 

Ans.  du  =  (a3  +  2a2x + 3ax2  +  4x3)dx, 

Ex.  2.  What  is  the  differential  of  the  function 
u  —  {a-\-  bx)2(?n  +  nx)3  ? 
Ans.  du  =  Sn(a  +  bx)2{ni  +  nx)2dx + 2b(a  +  bx)(?n  +  nx)3dx. 

Ex.  3.  What  is  the  differential  of  the  function 

u^ia  +  bx^Xc+exyi 
Ans.  da — 20(a + bx2)3(c + exi)iex3dx+6b(c+exif(a + bx2)2xdx. 

Ex.  4.  What  is  the  differential  of  the  function 
a  —  (a-\-Vx)3l 

.         ,       3(a-{-Vx)2dx 

A71S.  ail  — -zl . 

2Vx 


Examples    for  Practice.  275 

Ex.  5.  What  is  the  differential  of  the  function 


U: 


.x(a2  +  x2)Va2—x2 ? 


.       (a4  +  cV-4*4>^ 
Ans.  du  =  - 


Va2—x2 
Ex.  6.  Wha^s  the  differential  of  the  function 


_  a+x   ? 


Va— x 

Ans.  au  = 

2{a—xf 


ns.  au= -.ax. 


Ex.  7.  What  is  the  differential  of  the  function 

x 


U-- 


z+Vl-x2 

dx 


Ans.  du-. 


u  =  a  +  — — g? 


Vl—x2  +  2x(l—x2) 

Ex.  8.  What  is  the  differential  of  the  function 

4-Vx 

3+x2 

.         7        6(1—  x2)dx 
Ans.  du=y-1 '  ,  . 

Ex.  9.  What  is  the  differential  of  the  function 

a2—x2       „ 

u=— =-= r : 

ar+crar+ar 

.         7       —  2x(  2a4  +  2a2x2 — x*)dx 

Ans.  du  = ;  y     . 

(a4  +  aV'+a?4)2 

Ex.  10.  What  is  the  differential  of  the  function 

xn      7 


W  = 


A?2S.  C?W: 


(l+a?)n+1 

Ex.  11.  What  is  the  differential  of  the  function 
u  =  (a— x)y/ a2-{-x2  ? 

{a2— ax-\-2x2)dx 


Ans.  dn  = 


Va2  +  x2 
Ex.  12.  What  is  the  differential  of  the  function 


u  =  (a2 — x2)  Va  +  x  ? 


Ans.  du=\(a  —  5x)Va+x.dx. 


276  Examples    for   Practice. 

Ex.  13.  What  is  the  differential  of  the  function 

Vx2-\-y2 

7           axydx—ax2du 
Ans.  du= ^— ^-. 

%{x2  +  y2)2 

Ex.  14.  What  is  the  differential  of  the  function 

u  =  (2a2  +  3x2)(a2-x2fl 

Ans.  dn=  — 15x3V  a2— x2.dx. 

Ex.  15.  What  is  the  differential  of  the  function 


u~ 


Va  +  x+Va—x  „ 
Va  +  x—Va—x 


,  a2-\-aVdz— x2  7 

Ans.  du  = ...    dx. 

x2vdz—x2 

Ex.  16.  What  is  the  differential  of  the  function 

a-\-2bx  ? 

U~(a-\-bxf  ' 

.         7        —  2b2xdx 

Ans.  du=- — T. 

(a  +  bxy 

Ex.  17.  What  is  the  differential  of  the  function 
u=x  log-,  xl 

Ans.  dn  =  (l  -{-log.  x)dx. 

Ex.  18.  What  is  the  differential  of  the  function 

log.  x . 

x 

.         7       (1—  log.  x)dx 

Ans.  du—- % — - — . 

x* 

Ex.  19.  What  is  the  differential  of  the  function 

x 


log.  X  ' 

7       (log.  x—  l)dx 

Ans.  du  —  - — 7; ~ — . 

(log.  xy 

Ex.  20.  What  is  the  differential  of  the  function 

u  =  (\og.x)n? 

7       7?(log.  x)n~ldx 
Ajis.  du  —  -±—£ — - 


Examples   for   Practice.  277 

Ex.  21.  What  is  the  differential  of  the  function 
u=\og.  [a?+  Vx2  +  a2]  ? 

Ans.  du=- 


Vcc2  +  ( 
Ex.  22.  What  is  the  differential  of  the  function 

*  a+ V«  +#   ' 

.         ,  adx 

Ans.  au=- 


iVci2-\-l 


Ex.  23.  What  is  the  differential  of  the  function 

1 


c  VI -fa?2— a?  ' 


AftS.  o'tt: 


Ex.  24.  What  is  the  differential  of  the  function 


.        <  y/«+a?4-  Va— x  }  0 

;=lo°-  t  "7^= — 7=  5  ■ 
v  Va-fa?—  va— a? 


Ans.  du  = 


wVa2—x2 
Ex.  25.  What  is  the  differential  of  the  function 

°   (  Va-  V*  > 

,  -y/a.dx 

A?is.  au  =  -. — ;-. 

(a—x)yx 

Development  into  Series. 

Ex.  26.  Develop  into  a  series  the  function 
u=Va2-{-x2. 

Ans.  u  —  a-\ -A -— ,  etc. 

2a     2.4a3  ^  2.4.6a5     ' 

Ex.  27.  Develop  into  a  series  the  function 
u=V2x—  1. 

Aws.  u=y  —  HI—  a; .  etc.|. 

'  2      2  5 


278  Examples    for   Practice. 

Ex.  28.  Develop  into  a  series  the  function 

1 


Vb2  — x2 

Ans.  u  =  b~l  + -b~3x2 + -^-rb~5xi  +   '  '  -6~7a?6  + ,  etc. 
2  2.4  2.4.6 

Ex.  29.  Develop  into  a  series  the  function 

u=(a2 +  x2)3 '. 
lo     54        5.2     2        5.2.1  _s 

Arcs.  ^/;  =  a  J  +  oa  ^  +77^        ~ o~«aa    ■£+,  etc. 
o  0.0  0.0. y 

Ex.  30.  Develop  into  a  series  the  function 

1 


U  — 


-Vtf+i 


1      «*_       5a-8  5.9a?12  5.9.13a?16 

M_a_4a5+4^~4.8.12a13+4.8.12.16a"~' 

Ex.  31.  Develop  into  a  series  the  function 

i 

Z£  =  (a5-f-a4a?— a5)5. 

k      4    f       4.9     a?3       4.9.14      a4 
A,5.  ^^-ng^+gyas-  w  •l.2.3.4+>etc- 


Maxima  and  Minima. 

Ex.  32.  Fmd  the  values  of  a?  which  will  render  ^^  a  maximum 
or  a  minimum  in  the  equation 

u = x4  —  8x3 + 22a2 — 24a:  + 1 2. 
Ans.  This  function  has  a  maximum  value  when  x~2,  and 

a  minimum  value  when  a=l  or  3. 
Ex.  33.  Determine  the  maxima  and  minima  values  of  the 

function 

a2x 

\a—xy 
Ans.  u  has  a  maximum  when  ac=-\-a}  and  a  minimum 

when  as=—a. 
Ex.  34.  Find  the  values  of  x  which  will  render  the  function 
u  =  3a2x3— b*x+c5 
a  maximum  or  a  minimum. 

b2 
Ans.  There  is  a  maximum  corresponding  to  x=i—  — ,  and 

b2 

a  minimum  corresponding  to  x~  +^-. 


Examples   for   Practice.  279 

Ex.  35.  Find  the  values  of  x  which  will  render  u  a  maximum 
or  a  minimum  in  the  equation 

u  =  3a:4  —  1 6a:3  +■  6ar  +  72a:  -  1 . 
Ans.  This  function  has  a  maximum  value  when  x—-\-2, 

and  a  minimum  value  when  x~  —  1  or  +3. 
Ex.  36.  It  is  required  to  find  the  fraction  that  exceeds  its 
cube  by  the  greatest  possible  quantity. 

Ans.  +VJ. 
Ex.  37.  It  is  required  to  inscribe  the  greatest  rectangle  in  an 
ellipse  whose  axes  are  2 A  and  2B. 

Ans.  The  sides  of  the  rectangle  are  A-\/2  and  B-y/2. 
Ex.  38.  The  equation  of  a  certain  curve  is 
a2y  =  ax2  — x3. 
Required  its  greatest  and  least  ordinates. 

Ans.  When  o?=fa,  y  is  a  maximum;  when  x—Q,  y  is  a 

minimum. 
Ex.  39.  It  is  required  to  circumscribe  about  a  given  parabola 
an  isosceles  triangle  whose  area  shall  be  a  minimum.         , 
Ans.  The  altitude  of  the  triangle  is  four  thirds  of  the  axis 

of  the  parabola. 
Ex.  40.  Required  the  least  parabola  which  shall  circumscribe 
a  circle  whose  radius  is  R. 

Ans.  The  axis  of  the  parabola  is  fR,  and  its  base  is  3R. 
Ex.  41.  What  is  the  altitude  of  the  maximum  cylinder  which 
can  be  inscribed  in  a  given  paraboloid  ? 

Note. — A  paraboloid  is  a  solid  formed  by  the  revolution  of  a  parabola  about 
its  axis. 

Ans.  Half  the  axis  of  the  paraboloid. 
Ex.  42.  What  is  the  diameter  of  a  ball  which,  being  let  fall 
into  a  conical  glass  full  of  water,  shall  expel  the  most  water  pos- 
sible from  the  glass,  the  depth  of  the  glass  being  6  inches,  and 
its  diameter  at  top  5  inches  ? 

Ans.  4-^g-  inches. 

Subtangents  and  Subnormals. 

Ex.  43.  Find  the  value  of  the  subnormal  of  the  curve  whose 
equation  is 

y2  =  2a2  log.  a?. 

A         ^ 

Ans.  — . 


. 


280  Examples   for  Practice. 

Ex.  44.  Find  the  value  of  the  subnormal  of  the  curve  whose 

equation  is 

3flj/2  +  03  =  2#3. 

Ans.  — . 
a 

Ex.  45.  Required  the  subtangent  of  the  curve  whose  equa- 
tion is 

9        x3 

y  — — • 

a— x 

2x(a—x) 
Ans.  — — — — . 
3a— 2x 

Ex.  46.  Required  the  subtangent  of  the  curve  whose  equa- 
tion is 

xy2  =  a\a— x). 

2(ax—x2) 

Ans. * -. 

a 

Ex.  47.   Determine  when  the  subtangent  of  the  preceding 

curve  is  a  minimum. 

Ans.  When  x—\a. 

Ex.  48.  Find  the  value  of  the  subtangent  of  the  curve  whose 

equation  is 

x2y2  —  (a + x)2(b2 — x2). 

x(a-\-x)(b2—x2) 

Ans. i rr~u — -- 

x6  +  ao* 

Curvature  and  Curve  Lines. 
Ex.  49.  Determine  the  radius  of  curvature  at  any  point  of 
the  cubical  parabola  whose  equation  is 

yz  =  ax. 

Ans.  R=(V+< 

6a2y 

Ex.  50.   Determine  when  the  curvature  of  the  preceding 
curve  is  greatest. 

4/^2" 

Ans.  When  v  =  \/ — • 

J       v  45 

Ex.  51.  Determine  the  radius  of  curvature  at  any  point  of  the 
logarithmic  curve  whose  equation  is 

y  =  ax. 

3  J 

(M2+«2P 

Ans.  R  = y  ;  ,  M  being  the  modulus,  and  a  the  base. 


Examples    for   Practice.  281 

Ex.  52.   Determine  the  point  ot  greatest  curvature  of  the 
logarithmic  curve. 

m.  •  •  M 

Ans.  The  point  whose  ordinate  is  equal  to  —7-. 

Ex.  53.  Determine  whether  the  curve  whose  equation  is 

y3=x5 
has  a  point  of  inflection. 

Ans.  This  curve  has  a  point  of  inflection  at  the  origin. 
Ex.  54.  Determine  the  point  of  inflection  in  the  curve  whose 
equation  is 

ax2  =  a2y-\-x2y. 
Ans.  There  is  an  inflection  at  the  point  where  y  =  \a. 
Ex.  55.  Determine  the  point  of  inflection  in  the  curve  whose 
equation  is 

x2y2  =  a2(ax — x2). 

Ans.  There  is  a  point  of  inflection  corresponding  to  each 

r.         :  *  3a       ,  a 

01  the  points  x=—,  and  y=  ——/-•  A 

Ex.  56.  Determine  whether  the  curve  whose  equation  is 
(y  —  b)3  =  (x—a)2 
has  a  cusp  at  the  point  where  the  tangent  is  parallel  to  the  axis 
ofY. 


INTEGRAL  CALCULUS. 


Integration  of  Differentials. 
e  di 
dx 


Ex.  1.  Find  the  integral  of  the  differential 


du= 


(a—xf 

Ans.  u=: 


1 


"4(a— a?)4" 

Ex.  2.  Find  the  integral  of  the  differential 

Axdx 


dll: 


\\-x2f 


2 

Ans.  u  =  - ^  +  C. 

1  —x* 


Ex.  3.  Find  the  integral  of  the  differential 

,  2adx 

du  =  — - — 

xV  2ax— x2 


.                   2V2ax~x2     „ 
Ans.  u= f-C. 


282  Examples    for    Practice. 

Ex.  4.  Find  the  integral  of  the  differential 

7            xdx 
du = 3  • 

(2ax— x2)2  

Ans.  u=-\/  — t-C. 

a  v   2a— x 

Ex.  5.  Find  the  integral  of  the  differential 

7  x8dx 

du=- 


Va?  +  6x9 
A 

Ex.  6.  Find  the  integral  of  the  differential 

7  dx 

au= 


Va9  +  Gx9     c, 
Ans.  u— — |-U 

27 


VT+ 


X~ 


x3    ,    3X5        3.  bx1  r 

Ans.  w=*__+— -^-^+,etc.,  +C. 

Ex.  7.  Find  trie  integral  of  the  differential 

a;3  +  cr 

Ans.  w=log.  (#3  +  a2). 

Ex.  8.  Find  the  integral  of  the  differential 

5x3dx 


du  — 


3*4  +  7' 

Arcs.  w  =  fUog.(3^4  +  7). 


Ex.  9.  Find  the  integral  of  the  differential 

<fa  =    3  ,     a  , TT^- 

Ans.  i<=log.  (;r3+#2+^+l)- 
Ex.  10.  Find  the  integral  of  the  differential 
du—xz(a-\-bx2)-dx. 

Ans.  „  =  (_^— J— -p— . 
Ex.  11.  Find  the  integral  of  the  differential 

5 

cZw=ar2(a+a?3)  3da\ 

3^3  +  2a 
Aws.  w== a* 

2a2a?(a+#3)T 


Examples   for   Practice.  283 

Ex.  12.  Find  the  integral  of  the  differential 
du = x\a2  +  x2ydx. 

Ans.  u=^a2+a?f(4a?-3a?). 

Ex.  13.  Find  the  integral  of  the  differential 

,  dx 

\  du— 

\  x2Va2-\-x2 

Vcr+x2 

Ans.  ii— = . 

arx 

Ex.  14.  Find  the  integral  of  the  differential 

,  xedx 

au=- 


Va2-\-x2 

x5   ,- ;     5a2  fo^dx 

Ans.  fc-y«"+r — —  /    /-^-— - 
6  6  J  Vfl+a 


Ex.  15.  Find  the  integral  of  the  differential 

,         xl0dx 
du  = 


Va2—x2 

co    LLJu 


9a2  r    xBdx        x°     

Ans.  u  —  -—  I  —— -.A/2     > 


Ex.  16.  Find  the  integral  of  the  differential 

7  x5dx 

du  =  - 


V2ax- 


■x" 


.  9a  C     xAdx  x^    . 


Ex.  17.  Find  the  integral  of  the  differential 
,        x5dx 


x*-\-a'J' 

.              x*     a2x2     a*.        .  . 
Ans.  u=- — -f-  log.  {a?+az). 


Ex.  18.  Find  the  integral  of  the  differential 

,         x3dx 
du  =  —- 

Vl-x2 


.  2  {*    xdx        r2 

Ans.  u=-  I  —  _     i/i 

V  Vl-x2     3V1 


' 


284  Examples   for   Practice. 

Ex.  19.  Find  the  integral  of  the  differential 

(III: 


Va  +  bx2 


Ex.  20.  Find  the  integral  of  the  differential 

du = x\a  +  bx3)'Jdx. 

3 


2x2(a  +  bx3y-i      4a   .  .   ,  a 


Ex.  21.  Find  the  integral  of  the  differential 


cZtt: 


Ans.  u—-——  log.  (a  +  bx). 


Ex.  22.  Find  the  integral  of  the  differential 

xdx 


dlt: 


(a+bxf 

\b^2b2){a  +  bxf 


Ex.  23.  Find  the  integral  of  the  differential 

x~^dx 


du-- 


a  -f  bx' 


1  x 

Ans.  u  —  -  W. 


a     to '  a-\-bx' 


Ex.  24.  Find  the  integral  of  the  differential 

dx 


du  — 


x(a-\-bx3)' 


1  r3 

Ans.  u= —  losr. 


3a     °'a-\-bx3' 

Ex.  25.  Find  the  integral  of  the  differential 

dx 


du  — 

^  +  6^+8' 


Ans.  M=-loof. . 

2     °  #+4 


Examples   for  Practice.  285 

Rectification,  Quadrature,  etc. 

Ex.  26.  Determine  the  length  of  the  curve  of  a  parabola  cut 
off  by  a  double  ordinate  to  the  axis  whose  length  is  12,  the  ab- 
scissa being  2. 

Arts.  12.8374. 

Ex.  27.  Determine  the  circumference  of  an  ellipse  whose 
two  axes  are  24  and  18  inches. 

Ans.  66.31056. 

Ex.  28.  Required  the  equation  of  the  curve  whose  area  is 
equal  to  twice  the  rectangle  of  its  co-ordinates. 

Ans.  The  equation  is  xif  —  a. 

Ex.  29.  Determine  the  area  of  the  logarithmic  curve. 

Ans.  s=M(y'—l). 

Ex.  30.  Determine  the  area  of  an  hyperbola  whose  base  is 
24  and  altitude  10,  the  transverse  axis  being  30. 

Ans.  151.68734. 

Ex.  31.  Determine  the  area  of  the  curve  whose  equation  is 

9     9.  9     °    i        <l         r\ 

aiy's—~aiar-\-xt=\). 

Ans.  -cr. 

o 

Ex.  32.  Determine  the  surface  of  an  ellipsoid  whose  axes 
are  50  and  40. 

Ans.  5882.6385. 

Ex.  33.  Determine  the  convex  surface  of  a  paraboloid  whose 
axis  is  20,  and  the  diameter  of  whose  base  is  60. 

Ans.  3848.451. 

Ex.  34.  Determine  the  volume  of  the  solid  generated  by  the 

revolution  of  the  logarithmic  curve  about  the  axis  of  X 

v     Mrr?/2 
Ans.  \  — — . 

Ex.  35.  Determine  the  volume  of  the  solid  generated  by  the 
revolution  of  the  curve  whose  equation  is 

x3 


!f3 


a—x 

3 


Ans.  V=tt« 3log .-—-—— -a2x. 


286  Examples   for   Practice. 

Ex.  36.  Determine  the  volume  of  a  paraboloid  whose  axis  is 
30,  and  the  diameter  of  its  base  40 

Arts.  18849.556. 

Ex.  37.  Determine  the  volume  of  a  parabolic  spindle  which 
is  generated  by  the  revolution  of  a  parabola  about  its  base  26, 

the  height  being  h. 

Tr     lGrrbh2 
Ans.  V  = — — — . 
15 

Ex.  38.  Determine  the  volume  of  a  parabolic  spindle  whose 
length  is  80,  and  whose  greatest  diameter  is  32. 

Ans.  34314.569. 


THE    END. 


I 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  OEPT. 

Renewed  books  are  subject  to  in^dmerecalL 


LD  21-100m-6,'56 
(B9311sl0)476 


General  Library     _ 
University  of  California 
Berkeley 


APR    21  W 


Y'C  22333 


UNIVERS(JY  OF  CALIFORNI^UgRARY 


** 


I 


ft 


I 


